Tree-Structured Graphs and Hypergraphs
by Andreas Brandstädt, University of Rostock, Germany, and
Feodor F. Dragan, Kent State University, Ohio, U.S.A.
Contents
1 Introduction: Graphs With Tree Structure, Related Graph Classes
and Algorithmic Implications
5
2 Basic Notions and Results
2.1
2.2
7
Basic Graph Notions and Results . . . . . . . . . . . . . . . . . . . . .
7
2.1.1
Line graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.1.2
Transitively orientable graphs . . . . . . . . . . . . . . . . . . .
10
Some Basic Hypergraph Notions and Properties . . . . . . . . . . . . .
10
3 Chordal Graphs, Subclasses and Variants
3.1
16
Chordal Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.1.1
Structural Properties of Chordal Graphs . . . . . . . . . . . . .
16
3.1.2
Algorithmic Problems for Chordal Graphs . . . . . . . . . . . .
21
3.1.3
Convexity in Chordal Graphs and Powers of Chordal Graphs . .
23
3.2
Split Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.3
Interval Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3.1
Characterizations of Interval Graphs . . . . . . . . . . . . . . .
26
3.3.2
Unit Interval Graphs . . . . . . . . . . . . . . . . . . . . . . . .
29
3.3.3
Generalizations and Variants of Interval Graphs and Related
Graph Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Chords and Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.4
4 α-Acyclic Hypergraphs and Their Duals
35
4.1
A Motivation from Relational Database Theory . . . . . . . . . . . . .
35
4.2
Hypergraph Parameters: Coloring, Packing and Covering . . . . . . . .
37
4.2.1
Hypergraph 2-Coloring . . . . . . . . . . . . . . . . . . . . . . .
37
4.2.2
Matchings and Transversals: The Kőnig Property . . . . . . . .
38
Tree Structure of α-Acyclic Hypergraphs . . . . . . . . . . . . . . . . .
39
4.3
1
4.4
Join Trees as Maximum Spanning Trees
. . . . . . . . . . . . . . . . .
41
4.5
Graham’s Algorithm, Running Intersection Property and Other Desirable Properties Equivalent to α-Acyclicity . . . . . . . . . . . . . . . .
43
5 Dually Chordal Graphs
5.1
48
Dually Chordal Graphs, Maximum Neighborhood Orderings and Hypertrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5.2
Bipartite Graphs, Hypertrees and Maximum Neighborhood Orderings .
51
5.3
Further Matrix Notions . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
6 Totally Balanced Hypergraphs and Matrices
57
6.1
Totally Balanced Hypergraphs Versus β-Acyclic Hypergraphs . . . . . .
57
6.2
Totally Balanced Matrices . . . . . . . . . . . . . . . . . . . . . . . . .
59
7 Strongly Chordal and Chordal Bipartite Graphs
7.1
7.2
61
Strongly Chordal Graphs . . . . . . . . . . . . . . . . . . . . . . . . . .
61
7.1.1
Elimination Orderings of Strongly Chordal Graphs . . . . . . .
61
7.1.2
Γ-Free Matrices and Strongly Chordal Graphs . . . . . . . . . .
64
7.1.3
Strongly Chordal Graphs as Sun-Free Chordal Graphs . . . . .
66
Chordal Bipartite Graphs . . . . . . . . . . . . . . . . . . . . . . . . .
70
7.2.1
Tree Structure of Chordal Bipartite Graphs . . . . . . . . . . .
70
7.2.2
Relationship between Strongly Chordal and Chordal Bipartite
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
7.2.3
Edge Elimination Orderings of Chordal Bipartite Graphs . . . .
73
7.2.4
Recognition and Applications of Chordal Bipartite Graphs . . .
75
8 Distance-Hereditary Graphs, Subclasses and γ-Acyclicity
76
8.1
γ-Acyclicity and Berge-Acyclicity . . . . . . . . . . . . . . . . . . . . .
76
8.2
Cographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
8.2.1
Cograph characterization . . . . . . . . . . . . . . . . . . . . . .
77
8.2.2
Tree Structure and Optimization on Cographs . . . . . . . . . .
78
8.3
Distance-Hereditary Graphs . . . . . . . . . . . . . . . . . . . . . . . .
80
8.4
Minimum Cardinality Steiner Tree Problem in Distance-Hereditary Graphs 86
8.5
Further Important Subclasses of Distance-Hereditary Graphs . . . . . .
88
8.5.1
88
Ptolemaic Graphs and Bipartite Distance-Hereditary Graphs . .
2
8.5.2
Block Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Tree Structure Decomposition of Graphs
9.1
89
91
Modular Decomposition of Graphs . . . . . . . . . . . . . . . . . . . .
91
9.1.1
Basic Module Properties . . . . . . . . . . . . . . . . . . . . . .
91
9.1.2
Modular Decomposition Theorem . . . . . . . . . . . . . . . . .
93
9.2
Prime extensions of subgraphs in prime graphs . . . . . . . . . . . . . .
94
9.3
Graph classes with simple prime graphs . . . . . . . . . . . . . . . . . .
97
9.3.1
P4 -Sparse Graphs . . . . . . . . . . . . . . . . . . . . . . . . . .
97
9.3.2
Structure of co-chair-free chordal graphs . . . . . . . . . . . . . 100
9.3.3
Structure of bull-free chordal graphs . . . . . . . . . . . . . . . 103
9.4
Split Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
9.5
Homogeneous Decomposition
9.6
Clique Separator Decomposition . . . . . . . . . . . . . . . . . . . . . . 110
. . . . . . . . . . . . . . . . . . . . . . . 107
10 Treewidth and Clique-Width of Graphs
115
10.1 Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.2 Clique-Width of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.2.1 Clique-Width of H-Free Bipartite Graphs . . . . . . . . . . . . 120
10.2.2 Clique-Width of H-Free Chordal Graphs . . . . . . . . . . . . . 121
10.2.3 Clique-Width of (H1 , . . . , Hk )-Free Graphs . . . . . . . . . . . . 124
10.2.4 Power-Bounded Clique-Width of Graph Classes . . . . . . . . . 124
10.3 NLC-Width and Rank-Width of Graphs . . . . . . . . . . . . . . . . . 125
11 Leaf Powers
126
11.1 Basic notions and results . . . . . . . . . . . . . . . . . . . . . . . . . . 126
11.2 Basic facts on leaf powers . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.2.1 Leaf powers are strongly chordal . . . . . . . . . . . . . . . . . . 127
11.2.2 Leaf powers and fixed tolerance NeST graphs are the same class 128
11.2.3 Leaf rank and unit interval graphs . . . . . . . . . . . . . . . . . 129
11.3 The inclusion structure of k-leaf power classes . . . . . . . . . . . . . . 130
11.4 Characterizations of 3-leaf powers, 4-leaf powers and distance-hereditary
5-leaf powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.4.1 Basic leaf powers and substitution of cliques . . . . . . . . . . . 131
3
11.4.2 Characterizations of 3-leaf powers . . . . . . . . . . . . . . . . . 132
11.4.3 New characterizations of squares of trees . . . . . . . . . . . . . 134
11.4.4 Structure of (basic) 4-leaf powers . . . . . . . . . . . . . . . . . 138
11.4.5 Structure of (basic) distance-hereditary 5-leaf powers . . . . . . 141
11.5 Variants and generalizations of leaf powers . . . . . . . . . . . . . . . . 143
11.5.1 On (k, ℓ)-leaf powers . . . . . . . . . . . . . . . . . . . . . . . . 143
11.5.2 Exact leaf powers . . . . . . . . . . . . . . . . . . . . . . . . . . 147
11.5.3 Simplicial powers of graphs . . . . . . . . . . . . . . . . . . . . 148
11.6 Pairwise compatibility graphs and leaf powers . . . . . . . . . . . . . . 151
12 Complexity of Some Problems on Tree-Structured Graph Classes
153
12.1 Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
12.2 Independent Set and Chromatic Number . . . . . . . . . . . . . . . . . 153
12.3 Hamiltonian Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
12.4 Dominating Set and Steiner Tree . . . . . . . . . . . . . . . . . . . . . 154
12.5 Maximum Induced Matchings . . . . . . . . . . . . . . . . . . . . . . . 154
12.5.1 MIM for hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . 156
12.6 Exact Cover, Efficient Domination, and Efficient Edge Domination . . . 157
12.6.1 Exact Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.6.2 Efficient Domination . . . . . . . . . . . . . . . . . . . . . . . . 157
12.6.3 Efficient Edge Domination . . . . . . . . . . . . . . . . . . . . . 159
12.7 Graph Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13 Metric Tree-Like Structures in Graphs
162
13.1 Tree-Breadth, Tree-Length and Tree-Stretch of Graphs . . . . . . . . . 162
13.2 Hyperbolicity of Graphs and Embedding Into Trees . . . . . . . . . . . 164
4
1
Introduction: Graphs With Tree Structure, Related Graph Classes and Algorithmic Implications
The aim of this book is to present various aspects of tree structure in graphs and hypergraphs and its algorithmic implications together with some important graph classes
having nice and useful tree structure. In particular, we describe the hypergraph background and the tree structure of chordal graphs and some graph classes which are
closely related to chordal graphs such as chordal bipartite graphs, dually chordal graphs
and strongly chordal graphs as well as important subclasses.
A graph is chordal if it does not contain chordless cycles with at least four vertices
as induced subgraphs. The study of chordal graphs goes back to [259], and the many
aspects of chordal graphs are described in surveys and monographs such as [47, 93,
236, 359] and others. The interest in chordal graphs and related classes comes from
applications in computer science, in particular, relational database schemes [203, 204],
matrix analysis, models in biology, statistics, and others. The tree structure of chordal
graphs is closely related to the famous concept of treewidth introduced by Robertson
and Seymour [398]; this concept appears also under the name of partial k-trees in [8, 10]
(see e.g., [49]). Treewidth plays a central role in algorithmic and complexity aspects
on graphs.
Chordal graphs appear in the literature under various names such as triangulated graphs
(Chapter 4 of [236]), rigid-circuit graphs, perfect elimination graphs and others. Most of
the applications are due to the tree structure of chordal graphs which can be described
in terms of so-called clique trees (arranging the maximal cliques of the graph in a tree).
The hypergraph-theoretical background of chordal graphs is given by α-acyclic hypergraphs which play an important role in the theory of relational database schemes.
Various desirable properties of such schemes can be expressed in terms of various levels of acyclicity of hypergraphs [203, 204]: Chordal graphs correspond to α-acyclic
hypergraphs, dually chordal graphs correspond to the dual hypergraphs of α-acyclic
hypergraphs, strongly chordal graphs correspond to β-acyclic hypergraphs (which are
equivalent to totally balanced hypergraphs), ptolemaic graphs correspond to γ-acyclic
hypergraphs, and block graphs correspond to Berge-acyclic hypergraphs. Actually,
tree structure of hypergraphs was captured as arboreal hypergraphs by Berge [32, 33];
a hypergraph is α-acyclic if and only its dual is arboreal.
One of the chapters is devoted to structural properties of strongly chordal graphs and
chordal bipartite graphs which are closely related to each other. Another chapter is
dealing with distance-hereditary graphs and important subclasses.
We discuss also another width parameter of graphs, namely clique-width, and its relationship to treewidth as well as its algorithmic applications. Very similar to treewidth,
it is known that whenever a problem is expressible in a certain kind of Monadic Second
Order Logic, and one deals with a class of graph whose clique-width is bounded by a
constant then the problem is efficiently solvable on this class. This is one of the main
5
reasons for the great interest in treewidth and clique-width of (special) graphs. In general, it is NP-hard to determine the clique-width of a graph, and for many important
graph classes, the clique-width is unbounded. For some interesting classes, however,
clique-width is bounded.
One of the chapters deals with the concept of leaf powers which is motivated by phylogeny and is based on distances of leaves in trees. Leaf powers form an important
subclass of strongly chordal graphs.
Finally, we discuss some other graph parameters, namely, tree-length and tree-breadth
of a graph, tree-distortion and tree-stretch of a graph, and Gromov’s hyperbolicity of
a graph. All these parameters try to capture and measure tree likeness of a graph
from a metric point of view. The smaller such a parameter is for a graph, the closer
the graph is to a tree metrically. Graphs for which such parameters are bounded by
small constants have many algorithmic advantages; they allow efficient approximate
solutions for a number of optimization problems. Note also that recent empirical and
theoretical work has suggested that many real-life complex networks and graphs arising
in Internet applications, in biological and social sciences, in chemistry and physics have
tree-like structures from a metric point of view.
A first version of this book appeared as Chapter 29 in [64].
6
2
2.1
Basic Notions and Results
Basic Graph Notions and Results
Throughout this book, let G = (V, E) be a finite undirected graph without self-loops
and multiple edges with vertex set V and edge set E, and let |V | = n, |E| = m. For
a vertex v ∈ V , let N(v) = {u | uv ∈ E} denote the (open) neighborhood of v in G,
and let N[v] = {v} ∪ {u | uv ∈ E} denote the closed neighborhood of v in G. Vertices
x and y are adjacent in G if xy ∈ E, otherwise they are nonadjacent. We also say
that adjacent vertices see each other, and nonadjacent vertices miss each other. Two
vertices x, y ∈ V are true twins if N[x] = N[y], i.e., x and y have the same neighbors
and are adjacent to each other. Two vertices are false twins if they have the same
neighbors and are nonadjacent to each other.
The degree degG (v) of a vertex v is the number of its neighbors, degG (v) = |N(v)|. If
N[v] = V , v is a universal vertex of G. A clique is a set of vertices which are mutually
adjacent. An independent set (or stable set) in G is a set of vertices which are mutually
nonadjacent. For graph G = (V, E), the complement graph G has the same vertex set
V , and two distinct vertices are adjacent in G if and only if they are nonadjacent in G.
For U ⊆ V , let G[U] denote the subgraph of G induced by U. We write G − U for
G[V \ U], and we write G − u for G − {u}. Throughout this book, all subgraphs are
understood to be induced subgraphs.
Let F denote a set of graphs. A graph G is F -free if none of its induced subgraphs is
in F .
A graph is connected (or 1-connected) if there is a path between every pair of distinct
vertices. The maximal connected subgraphs are the connected components of G. U ⊆ V
is a cutset in G if G−U has more connected components than G. A k-cut in a connected
graph is a cutset with k vertices; a 1-cut is also called a cut-vertex. G is k-connected
if it has no cutset with at most k − 1 vertices. For a positive integer k, a k-connected
component in a graph G is a maximal (induced) k-connected subgraph of G; the 1connected components of G are the usual connected components, and the 2-connected
components of G are also called blocks of G.
For graphs G1 = (V1 , E1 ), G2 = (V2 , E2 ), the graph G1 ∩ G2 (G1 ∪ G2 , respectively) has
vertex set V1 ∩ V2 (V1 ∪ V2 , respectively) and edge set E1 ∩ E2 (E1 ∪ E2 , respectively).
Let dG (x, y) (or d(x, y) for short if G is understood) be the length, i.e., number of
edges, of a shortest path in G between x and y. For k ≥ 1, let Gk = (V, E k ) with
xy ∈ E k if and only if dG (x, y) ≤ k denote the k-th power of G. If G = H k then H is
a k-th root of G; if k = 2 then H is called a square root of G.
For k ≥ 1, let Pk denote a chordless path with k vertices and k −1 edges, and for k ≥ 4,
let Ck denote a chordless cycle with k vertices and k edges. A hole is a chordless cycle
Ck with k ≥ 5, and an odd hole is a chordless cycle C2k+1 with k ≥ 2. An antihole is
the complement graph of a hole, and an odd antihole is the complement graph of an
odd hole.
7
Definition 2.1 Let G be a finite undirected graph.
(i) G is chordal if for every k ≥ 4, G is Ck -free.
(ii) G is weakly chordal if for every k ≥ 5, G and G are Ck -free.
Clearly, every chordal graph is weakly chordal since C5 is isomorphic to C5 and Ck
contains C4 for every k ≥ 6.
A complete bipartite graph with r vertices in one color class and s vertices in the other
color class is denoted by Kr,s ; K1,3 is also called claw. Let Sk denote the (complete) sun
with 2k vertices u1 , . . . , uk and w1 , . . . , wk such that u1, . . . , uk is a clique, w1 , . . . , wk
is a stable set and for all i ∈ {1, . . . , k}, N(wi ) = {ui , ui+1} (index arithmetic modulo
k).
The diamond has four vertices and exactly one pair of nonadjacent vertices, i.e., it is a
K4 minus one edge, denoted K4− . In general, for k ≥ 4, a clique with k vertices minus
an edge is denoted Kk− ; it is the (k − 2)-th power of the induced path Pk . The gem (see
Figure 20) has five vertices such that four of them induce a P4 and the fifth is adjacent
to all of them.
Let O = {o1 , . . . , on } be a set of objects for which the notion of intersection makes
sense. A graph G = (V, E) with V = {v1 , . . . , vn } is the intersection graph of O if for
i 6= j, vi vj ∈ E if and only if oi ∩ oj 6= ∅. Examples of such classes are chordal graphs
and interval graphs as we will see in section 3.
A vertex set I is independent (or stable) in G if the vertices in I are pairwise nonadjacent. The independence number α(G) is the maximum cardinality of an independent
vertex set in G. Correspondingly, a vertex set is a clique in G if it is an independent
set in G. The clique number ω(G) is the maximum cardinality of a clique in G, i.e.,
ω(G) = α(G).
An assignment of labels (called colors) to vertices of G = (V, E) is an admissible
vertex coloring (vertex coloring for short) of G if no two vertices of G share the same
label. It can be interpreted as a partition of V into independent vertex set in G. The
chromatic number χ(G) is the minimum number of colors for any vertex coloring of G.
Correspondingly, the clique cover number κ(G) of G is minimum number of cliques for
any clique partition of G, i.e., κ(G) = χ(G).
The coloring problem of a graph is how to assign a minimum number of colors to the
vertices such that adjacent vertices get different colors. The chromatic number χ(G) of
the graph G is the minimum number of colors needed to color G. Obviously, for every
graph G, ω(G) ≤ χ(G) holds.
A graph is called χ-perfect if for every induced subgraph G′ of G (including G itself),
ω(G′) = χ(G′ ) holds. Let κ(G) = χ(G). Obviously, α(G) ≤ κ(G) holds.
A graph is called κ-perfect if for every induced subgraph G′ of G (including G itself),
α(G′ ) = κ(G′ ) holds. The following theorem is a celebrated result by Laszló Lovász
(see e.g. [93]):
8
Theorem 2.1 (Perfect Graph Theorem) A graph is χ-perfect if and only if it is
κ-perfect.
Now, G is called perfect if G is χ-perfect (κ-perfect).
Corollary 2.1 A graph G is perfect if and only if its complement graph G is perfect.
The famous Strong Perfect Graph Theorem was shown by Chudnovsky, Robertson,
Seymour and Thomas in [143]:
Theorem 2.2 (Strong Perfect Graph Theorem) A graph is perfect if and only if
it is odd-hole-free and odd-antihole-free.
See e.g. [93] for many important subclasses of perfect graphs; one of them is the class
of weakly chordal graphs [267].
Various graph parameters such as α(G), ω(G), χ(G), and κ(G) are hard to determine;
the corresponding decision problems are NP-complete (see e.g. [224]).
A matched co-bipartite graph is a co-bipartite graph, consisting of two disjoint cliques
X and Y of the same size and a perfect matching between them.
A vertex z ∈ V \ {x, y} distinguishes two vertices x, y ∈ V if z is adjacent to exactly
one of them, say zx ∈ E and zy ∈
/ E. A vertex subset U ⊆ V is a module in G if no
vertex from V \ U distinguishes two vertices in U. A nontrivial module is a module
with at least two but not all vertices. A nontrivial module of a graph is maximal if
there is no other nontrivial module of the graph containing it. A clique module in G is
a module which induces a clique in G. Distinct vertices x, y ∈ V are true twins in G if
N[x] = N[y]. They are false twins if N(x) = N(y), i.e., they have the same neighbors
and are nonadjacent to each other.
1
0 ,
Disjoint vertex sets X, Y form a join (cojoin, respectively), denoted by X Y
(X Y
respectively), if for all pairs x ∈ X, y ∈ Y , xy ∈ E (xy 6∈ E, respectively) holds.
2.1.1
Line graphs
Let G = (V, E) be a graph. The line graph of G is the graph L(G) = (E, E ′ ) with
ee′ ∈ E ′ if and only if e 6= e′ and e ∩ e′ 6= ∅.
Theorem 2.3 (Beineke [30]) A graph is a line graph if and only if it does not contain
any induced subgraph isomorphic to one of the graphs in Figure 1.
See e.g. [359] for other characterizations of line graphs.
9
Figure 1: The forbidden induced subgraphs for line graphs
2.1.2
Transitively orientable graphs
A graph G = (V, E) is called transitively orientable if its edge set E can be oriented
as E ′ in such a way that for all oriented edges (x, y), (y, z) ∈ E ′ , (x, z) ∈ E ′ holds.
+++++++++++++
2.2
Some Basic Hypergraph Notions and Properties
A pair H = (V, E) is a (finite) hypergraph if V is a finite vertex set and E is a collection
of subsets of V (the edges or hyperedges of H). Hypergraphs are a natural generalization
of undirected graphs; unlike edges, hyperedges are not necessarily two-elementary. In
many cases, hyperedges containing exactly one vertex (so-called loops) are excluded.
Equivalently, a hypergraph H = (V, E) with V = {v1 , . . . , vn } and E = {e1 , . . . , em }
can be described by its n × m vertex-hyperedge incidence matrix M(H) with entries
mij ∈ {0, 1} and mij = 1 ⇐⇒ vi ∈ ej for i ∈ {1, . . . , n} and j ∈ {1, . . . , m}.
Subsequently, we collect some basic notions and properties - see e.g. [33].
Definition 2.2 A hypergraph H = (V, E) is simple if it has no repeated edges. Moreover, if no hyperedge e ∈ E is properly contained in another hyperedge e′ ∈ E then H
is called a Sperner family or clutter.
In the database community (see e.g. [28]), clutters are called reduced hypergraphs.
Definition 2.3 Let H = (V, E) be a finite hypergraph.
(i) The subhypergraph induced by the subset A ⊆ V is the hypergraph H[A] =
(A, EA ) with edge set EA = {e ∩ A | e ∈ E}.
10
(ii) The partial hypergraph
given by the edge subset E ′ ⊆ E is the hypergraph with
S ′
the vertex set E and the edge set E ′ .
Note that both restrictions A ⊂ V and E ′ ⊂ E can be combined in a subhypergraph
H ′ [A] = (A, EA′ ) with edge set EA′ = {e ∩ A | e ∈ E ′ ⊂ E} called partial subhypergraph
in [33].
The partial hypergraphs [33] are called subhypergraphs in [203]. Since this may cause
confusion, we also use the name edge-subhypergraphs for partial hypergraphs and vertexsubhypergraphs in case (i).
Dualization is a classical concept which is well known from geometry; there, points and
hyperplanes exchange their role. Here, dualization means that vertices and hyperedges
exchange their role:
Definition 2.4 Let H = (V, E) be a finite hypergraph. For v ∈ V , let Ev = {e ∈ E |
v ∈ e}. The dual hypergraph H ∗ = (E, E ∗) of H has vertex set E and hyperedge set
E ∗ = {Ev | v ∈ V }.
If the hypergraph H is given in terms of its incidence matrix M(H) then the incidence
matrix of the dual of H is the transposal of M(H): M(H ∗ ) = (M(H))T .
Evidently, the dual of the dual of H is isomorphic to H itself since for the twofold
transposal, ((M(H))T )T = M holds. Thus:
Proposition 2.1 (H ∗)∗ ∼ H.
Graphs and hypergraphs are closely related to each other. The next definition represents two examples.
Definition 2.5 Let H = (V, E) be a finite hypergraph.
(i) The 2-section graph 2SEC(H) of H has the vertex set V , and two vertices u, v
are adjacent if u and v are contained in a common hyperedge: ∃e ∈ E such that
u, v ∈ e.
(ii) The line graph L(H) = (E, F ) is the intersection graph of E, i.e., for any e, e′ ∈ E
with e 6= e′ , ee′ ∈ F ⇐⇒ e ∩ e′ 6= ∅.
The 2-section graph of H is denoted by [H]2 in [33]; the line graph is also called
representative graph in [33]. Again, these notions have different names in different
communities; the 2-section graph is also called adjacency graph in [246], primal graph
[247] or Gaifman graph [222] and has no name but is denoted by G(H) in [28]. The
line graph is also called dual graph in [176, 352].
The following isomorphism is easy to see:
11
Proposition 2.2 2SEC(H) ∼ L(H ∗ ).
A subfamily E ′ ⊆ E is called pairwise intersecting if for all e, e′ ∈ E ′, e ∩ e′ 6= ∅.
Definition 2.6 Let H = (V, E) be a hypergraph.
(i) H is conformal if every clique C in 2SEC(H) is contained in a hyperedge e ∈ E.
(ii) H has the Helly property if Tevery pairwise intersecting subfamily E ′ ⊆ E has
nonempty total intersection: E ′ 6= ∅.
The following is easy to see:
Proposition 2.3 H is conformal if and only if H ∗ has the Helly property.
Proof. “=⇒”: Assume that H is conformal, i.e., every clique Q in in 2SEC(H) is
contained in a hyperedge eQ ∈ E. We show that H ∗ has the Helly property: Let V ′ ⊆ V
be a family of vertices such that for all x, y ∈ V ′ , Ex ∩ Ey 6= ∅. Then V ′ is a clique
in H, T
and since H is conformal, there is a hyperedge eV ′ such that V ′ ⊆ eV ′ . Now
eV ′ ∈ x∈V ′ Ex and thus, H ∗ has the Helly property.
“⇐=”: Assume that H ∗ = (E, {Ev | v ∈ V }) has the Helly property. We show that
H is conformal: Let Q be a clique in 2SEC(H). Then for all x, y ∈ Q, there is a
hyperedge ex,y ∈ ETcontaining both of them, i.e., Ex ∩ Ey 6= ∅. Since H ∗ has the Helly
property, we have x∈Q Ex 6= ∅. Thus, there is a hyperedge eQ ∈ E with Q ⊆ eQ .
2
The next theorem gives a polynomial time criterion for testing the Helly property of
a hypergraph. It is closely related to an earlier criterion for conformality given by
Gilmore which will be mentioned in Theorem 2.5.
For a hypergraph H = (V, E) and for any 3-elementary set A = {a1 , a2 , a3 } ⊆ V , let
EA denote the set of all hyperedges e ∈ E such that |e ∩ A| ≥ 2.
Theorem 2.4 (Ryser [406], Berge, Duchet; see [33]) A hypergraph H = (V, E)
has the Helly property if and only if for all 3-elementary sets A = {a1 , a2 , a3 } ⊆ V , the
total
T intersection of all hyperedges containing at least two vertices of A is nonempty:
EA 6= ∅.
Proof. “=⇒”: Let H be a hypergraph with the Helly property, let A = {a1 ,T
a2 , a3 } ⊆
V , and let EA be as above. Then for all ei , ej ∈ EA , ei ∩ ej 6= ∅ and thus, EA 6= ∅
since H has the Helly property.
“⇐=”: Now assume that {e1 , . . . , eℓ } ⊆ E is a collection of pairwise intersecting hyperedges. If ℓ = 2 then obviously their total intersection is nonempty; thus let ℓ > 2.
We assume inductively that the assertion of nonempty total intersection is true for less
than ℓ hyperedges with pairwise nonempty intersection.
12
Then by the induction hypothesis, e1 ∩ . . . ∩ eℓ−1 6= ∅; let a1 ∈ e1 ∩ . . . ∩ eℓ−1 . Moreover,
e2 ∩ . . . ∩ eℓ 6= ∅; let a2 ∈ e2 ∩ . . . ∩ eℓ . Finally e1 ∩ eℓ 6= ∅; let a3 ∈ e1 ∩ eℓ .
Let A := {a1 , a2 , a3 }. It is easy to see that in the case |A| < 3 we are done. Thus
let |A| = 3; every ei , i = 1, . . . , ℓ, contains at least two elements from A, and by
assumption, e1 ∩ . . . ∩ eℓ 6= ∅.
2
An obvious consequence of Theorem 2.4 is:
Corollary 2.2 Testing the Helly property for a given hypergraph can be done in polynomial time.
Corollary 2.3 Every collection of subtrees of a tree has the Helly property.
Proof. Let T be a tree with at least three vertices (otherwise the assertion is obviously
fulfilled), and let a, b, c be any three vertices in T . We consider the set of all subtrees
of T containing at least two of the vertices a, b, c. Let P (x, y) denote the uniquely
determined path in the tree T between x and y. Let x0 denote the last vertex in
P (a, b) ∩ P (b, c) (this intersection contains at least vertex b). Then P (a, c) consists of
P (a, x0 ) followed by P (x0 , c). Thus the three paths P (a, b), P (b, c) and P (a, c) have
vertex x0 in common, i.e., x0 is contained in every subtree of T which contains at least
two of the vertices a, b, c. Thus, by Theorem 2.4, every system of subtrees has the Helly
property.
2
A nice inductive proof of Corollary 2.3 is given in a script by Alexander Schrijver:
The induction is on |V (T )|. If |V (T )| = 1 then the assertion is trivial. Now assume
|V (T )| ≥ 2, and let S be a collection of pairwise intersecting subtrees of T . Let t be
a leaf of T . If there exists a subtree of T consisting only of t, the assertion is trivial.
Hence we may assume that each subtree in S containing t also contains the neighbor
of t in T . So, after deleting t from T and from all subtrees in S, this collection is still
pairwise intersecting, and the assertion follows by induction.
Actually, Theorem 2.4 is formulated in a more general way in [33]; there are various
interesting generalizations of the Helly property.
According to Proposition 2.3, Theorem 2.4 can be dualized as follows:
Theorem 2.5 (Gilmore, see [33]) Let H = (V, E) be a hypergraph. H is conformal
if and only if for all 3-elementary edge sets A = {e1 , e2 , e3 } ⊆ E of hyperedges, there is
a hyperedge e ∈ E with (e1 ∩ e2 ) ∪ (e1 ∩ e3 ) ∪ (e2 ∩ e3 ) ⊆ e.
Proof. “=⇒”: Obviously, (e1 ∩ e2 ) ∪ (e1 ∩ e3 ) ∪ (e2 ∩ e3 ) is a clique in the 2-section
graph 2SEC(H) of H. By conformality, there is a hyperedge e with (e1 ∩ e2 ) ∪ (e1 ∩
e3 ) ∪ (e2 ∩ e3 ) ⊆ e.
“⇐=”: Let A = {e1 , e2 , e3 } ⊆ E and let Eu be a hyperedge in H ∗ containing at least
two of e1 , e2 , e3 . Then u ∈ (e1 ∩ e2 ) ∪ (e1 ∩ e3 ) ∪ (e2 ∩ e3 ) and thus also u ∈ e. Thus, e
is in the total intersection of all hyperedges Eu which contain at least two of e1 , e2 , e3 .
13
Then by Theorem 2.4, H ∗ has the Helly property and thus, by Proposition 2.3, H is
conformal.
2
There is a third type of graphs derived from a hypergraph H = (V, E), namely the
bipartite vertex-edge incidence graph I(H) (which is a reformulation of the incidence
matrix of H in terms of a bipartite graph). The two color classes of I(H) are the sets
V and E, respectively, and a vertex v and an edge e are adjacent if and only if v ∈ e.
More formally:
Definition 2.7 Let H = (V, E) be a finite hypergraph. In the bipartite incidence graph
I(H) = (V, E, I) of H, v ∈ V and e ∈ E are adjacent if and only if v ∈ e.
In the other direction, namely from graphs to hypergraphs, the most basic constructions
are the following:
Definition 2.8 Let G = (V, E) be a graph.
(i) The clique hypergraph C(G) consists of the ⊆-maximal cliques of G.
(ii) The neighborhood hypergraph N (G) consists of the closed neighborhoods N[v] of
all vertices v in G.
(iii) The disk hypergraph D(G) consists of the iterated closed neighborhoods N i [v],
i ≥ 1, of all vertices v in G, where N 1 [v] := N[v] and N i+1 [v] := N[N i [v]].
Note that in general, the neighborhood hypergraph N (G) is not simple since different
vertices can have the same closed neighborhood in G. The following is easy to see:
Proposition 2.4 N (G) is self-dual, i.e., (N (G))∗ ∼ N (G).
Moreover, the 2-section graph of C(G) is isomorphic to G and thus, C(G) is conformal.
Note that a hypergraph uniquely determines its 2-section graph but not vice versa.
Lemma 2.1 Every conformal Sperner hypergraph H = (V, E) is the clique hypergraph
of its 2-section graph 2SEC(H): H = C(2SEC(H)).
Proof. Let H be conformal and Sperner. We show:
1) For every e ∈ E, e is a maximal clique in 2SEC(H):
Obviously, e is a clique in 2SEC(H) and thus, there is a maximal clique C ′ in 2SEC(H)
with e ⊆ C ′ . Since H is conformal, there is an e′ ∈ E with C ′ ⊆ e′ , i.e., e ⊆ C ′ ⊆ e′
and since H is Sperner, e = C ′ = e′ follows.
2) For every maximal clique C in 2SEC(H), C ∈ E holds:
By conformality of H, there is e ∈ E with C ⊆ e, and since e is a clique in 2SEC(H),
there is a maximal clique C ′ in 2SEC(H) with e ⊆ C ′ , i.e., C ⊆ e ⊆ C ′ . By maximality
of C, C = e = C ′ follows.
2
14
Recall that the square G2 = (V, E 2 ) of a graph G is defined as xy ∈ E 2 for x 6= y if
and only if dG (x, y) ≤ 2 (see Section 2). The following is easy to see:
Proposition 2.5 G2 ∼ L(N (G)).
Definition 2.9 For graph G = (V, E), let B(G) = (V ′ , V ′′ , F ) denote the bipartite
graph with two disjoint copies V ′ and V ′′ of V , and for v ′ ∈ V ′ and w ′′ ∈ V ′′ , v ′ w ′′ ∈ F
if and only if either v = w or vw ∈ E.
The following is easy to see:
Proposition 2.6 B(G) ∼ I(N (G)).
The line graph of C(G) is the classical clique graph operator in graph theory:
Definition 2.10 Let G be a graph.
(i) The clique graph K(G) of G is defined as K(G) = L(C(G)).
(ii) G is a clique graph if there is a graph G′ such that G is the clique graph of G′ ,
i.e., G = K(G′ ).
Theorem 2.6 (Roberts, Spencer [397]) A graph G is a clique graph if and only if
some class of complete subgraphs of G covers all edges of G and has the Helly property.
See [93, 359] and in particular the survey [417] by Szwarcfiter for more details on clique
graphs. Recognizing whether a graph is a clique graph is NP-complete [5].
15
3
Chordal Graphs, Subclasses and Variants
In this section, we collect some notions and well-known facts on chordal graphs which
are described in various monographs; see e.g. [236] and the survey [93] as well as [359]
for details. In order to make this section self-contained, we briefly repeat some of the
basic definitions and properties. Throughout this section, let G = (V, E) be a finite
undirected graph which is simple (i.e., loop-free and without multiple edges).
3.1
3.1.1
Chordal Graphs
Structural Properties of Chordal Graphs
Recall Definition 2.1 for chordal graphs. Obviously, trees and forests are chordal since
they are cycle-free for any cycle length. Chordal graphs have a nice separator property
which was found by Dirac [183].
Definition 3.1
(i) The vertex set S ⊆ V is a separator (or cutset) for nonadjacent vertices a, b ∈ V
(a-b-separator) if a and b are in different connected components in G[V \ S].
(ii) S is a minimal a-b-separator if S is an a-b-separator and no proper subset of S
is an a-b-separator.
(iii) S is a (minimal) separator if there are vertices a, b such that S is a (minimal)
a-b-separator.
Theorem 3.1 (Dirac [183]) A graph G is chordal if and only if every minimal separator in G induces a clique.
Proof. “⇐=”: Let (a, x, b, y1 , y2, . . . , yk , a) be a simple cycle of G, k ≥ 1. Every cycle
of length ≥ 4 is of this form (if there is any cycle in G).
If ab ∈ E then the cycle contains a chord. If ab ∈
/ E then the vertices a and b can be
separated. Every minimal a-b-separator contains x and yi for some i, 1 ≤ i ≤ k, and
since the separator induces a clique, xyi ∈ E, thus the cycle contains a chord.
“=⇒”: Let S be a minimal a-b-separator and let G[A] (G[B], respectively) the connected component of G[V \ S] containing a (b, respectively). Since S is a minimal
separator, every x ∈ S is adjacent to a vertex in G[A] and to a vertex in G[B].
Thus, for each pair x, y ∈ S with x 6= y, there are paths P1 = (x, a1 , . . . , ar , y) and
P2 = (y, b1, . . . , bs , x), ai ∈ G[A], bj ∈ G[B], i ∈ {1, . . . , r}, j ∈ {1, . . . , s}. Without loss
of generality, choose such paths of minimal length. Then (x, a1 , . . . , ar , y, b1 , . . . , bs , x)
is a simple cycle of length ≥ 4. By assumption, this cycle must have a chord. Since,
moreover, the paths P1 , P2 have minimal length, the only possible chord is the edge
xy ∈ E. Thus, S induces a clique.
2
16
Definition 3.2
(i) A vertex v ∈ V is simplicial in G if N(v) induces a clique in G.
(ii) An ordering (v1 , . . . , vn ) of the vertices of V is a perfect elimination ordering
(p.e.o.) of G if for all i ∈ {1, . . . , n}, the vertex vi is simplicial in the remaining
subgraph Gi := G[{vi , . . . , vn }].
Obviously, the notion of a simplicial vertex generalizes leaves in trees.
Lemma 3.1 (Dirac [183]) Every chordal graph with at least one vertex contains a
simplicial vertex. If G is not a clique then G contains at least two nonadjacent simplicial
vertices.
Proof. Let G = (V, E) be a chordal graph. If G is a complete graph then the assertion
is trivially fulfilled. Now let a, b ∈ V , ab ∈
/ E, and, as induction hypothesis, let Lemma
3.1 be fulfilled for all graphs with less vertices than G. Let S be a minimal a-b-separator
with a ∈ G[A], b ∈ G[B], where G[A] (G[B], respectively) is the connected component
of G[V \ S] containing a (containing b, respectively). By Theorem 3.1, S is a clique.
By induction hypothesis, G[A ∪ S] either contains two nonadjacent simplicial vertices,
one of them being in A since S induces a clique, or G[A ∪ S] itself is a clique and every
vertex of A is simplicial in G[A ∪ S]. Since N(A) ⊆ A ∪ S, every vertex in A which
is simplicial in G[A ∪ S] is also simplicial in G. Analogously, G[B] contains a vertex
which is simplicial in G (and nonadjacent to any vertex in G[A]).
2
Corollary 3.1 (Dirac [183]; Fulkerson, Gross [221]) G is chordal if and only if
G has a perfect elimination ordering. Moreover, every simplicial vertex of a chordal
graph G can be the first vertex of a perfect elimination ordering of G.
Proof. “=⇒”: Let G be chordal. Then by Lemma 3.1, G contains a simplicial vertex
v. Since G[V \ {v}] is also chordal, the repeated deletion of simplicial vertices leads to
a perfect elimination ordering of G.
“⇐=”: Let (v1 , . . . , vn ) be a perfect elimination ordering of G and let C be a simple
cycle of G, where v is the leftmost vertex of C in the perfect elimination ordering.
Since |N(v) ∩ C| ≥ 2, the neighbors of v in C have a chord. Thus, G is chordal.
2
Let s1 = (a1 , . . . , ak ) and s2 = (b1 , . . . , bℓ ) be vectors of positive integers; such vectors
are used as vertex labels in Algorithm 3.1. Then s1 is lexicographically smaller than s2
(s1 < s2 ) if either there is an index i ≤ min(k, ℓ) such that ai < bi and aj = bj for all
j, 1 ≤ j ≤ i − 1 or k < ℓ and ai = bi for all i with 1 ≤ i ≤ k.
If s = (a1 , . . . , ak ) is a vector of positive integers and p is a positive integer then
s + p := (a1 , . . . , ak , p). Let () denote the empty label.
The subsequent algorithm Lexicographic Breadth-First Search (LexBFS) assigns a distinct number to each vertex of V depending on its label and, during the procedure,
updates labels of neighbors as follows:
17
Algorithm 3.1 (Lexicographic Breadth-First Search (LexBFS))
Input:
Output:
A graph G = (V, E)
A LexBFS ordering σ = (v1 , . . . , vn ) of V .
begin for all v ∈ V do l(v) := ();
for k := |V | = n downto 1 do
choose a vertex v with lexicographically largest label l(v);
set σ(k) := v;
for all u ∈ V ∩ N(v) do l(u) := l(u) + k;
set V := V \ {v};
endfor;
end
Theorem 3.2 (Rose, Tarjan, Lueker [404]) For a chordal graph G, every LexBFS
ordering of G is a p.e.o. of G.
This leads to a linear time recognition algorithm for chordality of graphs [404] as
follows:
1. For a given graph G = (V, E), determine a LexBFS ordering σ.
2. Check if σ is a p.e.o. of G; if yes, G is chordal, otherwise G is not chordal.
This can be done in linear time (see e.g. [236] for details); LexBFS as partition refinement explains why step 1 can be done in linear time.
Subsequently, for an ordering σ = (v1 , . . . , vn ) of V and two vertices x = vi and y = vj ,
x < y means that i < j. LexBFS orderings of graphs can be characterized by the
following 4-point condition:
For all a < b < c with ac ∈ E, bc ∈
/ E there is d > c with bd ∈ E and ad ∈
/ E.
(1)
Theorem 3.3 ([68]) For any graph G = (V, E), an ordering σ = (v1 , . . . , vn ) of V is
a LexBFS ordering of G if and only if σ fulfills the 4-point condition (1).
See [149] for a survey on LexBFS and other vertex orderings.
There is another linear-time algorithm for recognizing chordal graphs called Maximum
Cardinality Search (MCS) introduced in [421] (see also [236]):
Algorithm 3.2 (Maximum Cardinality Search (MCS))
Input:
Output:
A graph G = (V, E)
A MCS ordering σ = (v1 , . . . , vn ) of V .
18
begin for k := |V | = n downto 1 do
choose a vertex v with a maximal number of numbered neighbors;
set σ(k) := v; V := V \ {v};
endfor;
end
For a collection T of subtrees of a tree T , let the vertex intersection graph GT of T be
the graph having the elements of T as its vertices, and two subtrees t and t′ from T
are adjacent in GT if they share a vertex in T .
Proposition 3.1 The vertex intersection graph of a collection of subtrees in a tree is
chordal.
Proof. Let G = (V, E) be the vertex intersection graph of a collection of subtrees
in a tree T . Suppose G contains a chordless cycle (v0 , v1 , . . . , vk−1 , v0 ) with k > 3
corresponding to the sequence of subtrees T0 , T1 , . . . , Tk−1 , T0 of the tree T ; that is,
Ti ∩ Tj 6= ∅ if and only if i and j differ by at most one modulo k. All index arithmetic
will be done modulo k.
Choose a point ai from Ti ∩ Ti+1 (i = 0, . . . , k − 1). Let bi be the last common point on
the (unique) simple paths from ai to ai−1 and ai to ai+1 . These paths lie in Ti and Ti+1 ,
respectively, so that bi also lies in Ti and Ti+1 . Let Pi+1 be the simple path connecting
bi to bi+1 in T . Clearly Pi ⊆ Ti , so Pi ∩ Pj = ∅ for i and j differing
by more than 1
S
mod k. Moreover, Pi ∩ Pi+1 = {bi } for i = 0, . . . , k − 1. Thus, i Pi is a simple cycle
in T , contradicting the definition of a tree.
2
The tree structure of chordal graphs is described in terms of so-called clique trees of
the maximal cliques of the graph; see Theorem 3.4.
Definition 3.3 Let C(G) denote the family of ⊆-maximal cliques of graph G = (V, E).
A clique tree T of G has the maximal cliques of G as its nodes, and for every vertex v
of G, the maximal cliques containing v form a subtree of T .
This notion will be generalized in Chapter 4 on hypergraphs; it can be taken for
defining α-acyclicity of a hypergraph (see Definition 4.4). The existence of a clique
tree characterizes chordal graphs:
Theorem 3.4 (Buneman [113], Gavril [228], Walter [432]) A graph is chordal
if and only if it has a clique tree.
Proof. “⇐=”: Assume that G has a clique tree T . If T has only one node then G
is a clique and thus chordal. Now let T have k > 1 nodes and assume as induction
hypothesis that the assertion is true for clique trees with less than k nodes. Let C be a
leaf node in T , let C ′ be its neighbor in T , let VC be the subset of G vertices occuring
only in C, and let T ′ be the clique tree restricted to V \ VC .
19
VC must be nonempty since otherwise, C ⊂ C ′ which is impossible by maximality of
the cliques. Now start a p.e.o. of G with the vertices of VC and then continue with a
p.e.o. for G − VC which must exist since T ′ has less nodes than T .
“=⇒”: For this direction, we assume that G is chordal, and let σ = (v1 , . . . , vn ) be
a p.e.o. of G. We inductively assume that G′ := G[{v2 , . . . , vn }] has a clique tree T ′ .
Since v1 is simplicial in G, N(v1 ) is a clique but not necessarily a maximal one in G′ .
In the case that it is a maximal clique in G′ represented by the node Q in T ′ then
add v1 to Q, and now T ′ is a clique tree of G. In the other case when N(v1 ) is not a
maximal clique in G′ , it is contained in a maximal clique of G′ represented by node Q
in T ′ . Now add a new node N[v1 ] to T ′ and connect it to Q; obviously, this extension
of T ′ is a clique tree of G.
The following version described by Spinrad in [414] leads to linear time construction
of clique trees: We construct a clique tree for the subgraph Gi = G[vi , . . . , vn ] for all
vertices, starting with i = n and ending with i = 1. Let Ci be the clique consisting
of vi and all neighbors vj of vi , j > i. After each vertex vi is processed, vi is given a
pointer to the clique Ci in the tree. We note that vertices may be added to this clique
later in the algorithm, but vi will always point to a clique which contains Ci .
Let vi be the next vertex considered, and assume we know the clique tree on the graph
induced by vertices vi+1 , . . . , vn . We need to add Ci to the clique tree. Let vj be the
first (i.e., leftmost) vertex of Ci on the right of vi in σ. If |Ci | = |Cj | + 1, and the
clique pointed to by vj is equal to Cj then we add vi to this clique; in other words, Ci
replaces Cj in the tree. Otherwise, add Ci as a new node of the tree. Connect Ci to
the tree by adding an edge from Ci to the clique pointed to by vj .
To see that the algorithm is correct, it is sufficient to look at two cases. Either Cj is
a maximal clique in Gi+1 = G[{vi+1 , . . . , vn }] or it is not. If Cj is a maximal clique, it
clearly must be replaced by Ci if Cj is contained in Ci , which occurs if Ci = Cj ∪ {vi },
and the algorithm does this correctly. If Cj is not a maximal clique in Gi+1 or Ci does
not contain Cj , then Ci cannot contain any maximal clique of Gi+1 , and must be added
as a new node. All elements of Ci − vi are in the clique pointed to by vj , so the subtrees
generated by the occurrences of all vertices remain connected.
2
Since a p.e.o. of a chordal graph can be determined in linear time, the proof of Theorem
3.4 implies:
Theorem 3.5 Given a chordal graph G = (V, E), a clique tree of G can be constructed
in linear time O(|V | + |E|).
A consequence of Theorem 3.4 and Proposition 3.1 is:
Corollary 3.2 (Buneman [113], Gavril [228], Walter [432]) A graph is chordal
if and only if it is the intersection graph of certain subtrees of a tree.
Proof. “=⇒”: If G is chordal then by Theorem 3.4, it has a clique tree, say T .
Thus, for an edge xy in G, the subtree Tx of maximal cliques containing x, and the
corresponding subtree Ty intersect in a maximal clique Q containing x and y.
20
“⇐=”: If G is the intersection graph of certain subtrees of a tree then, by Proposition
3.1, G is chordal.
2
Interestingly, a clique tree of a chordal graph G gives also the minimal separators of
G:
Lemma 3.2 ([26, 274]) Let G = (VG , EG ) be a chordal graph with clique tree T =
(C(G), ET ). Then S ⊆ VG is a minimal separator in G if and only if there are maximal
cliques Qi , Qj of G with Qi Qj ∈ ET such that S = Qi ∩ Qj .
3.1.2
Algorithmic Problems for Chordal Graphs
The specific structure of chordal graphs and in particular the perfect elimination orderings allow to solve various problems efficiently which is well described in [236] for the
basic problems Maximum Clique, Maximum Independent Set, Chromatic Number and
Clique Covering. A good example is the use of p.e.o. for determining the chromatic
number χ(G) and the clique number ω(G) of a given chordal graph G; obviously, for
every graph, ω(G) ≤ χ(G).
Let G = (V, E) be a chordal graph with perfect elimination ordering σ = (v1 , . . . , vn )
of G. The following algorithm using σ from right to left colors the vertices of G with
ω(G) colors. Starting with vn , a color is assigned to every vi , i = n, n − 1, . . . , 2, 1, as
follows: Since vi is simplicial in Gi = G[{vi , . . . , vn )}], it has at most ω(G)−1 neighbors
in Gi (which, as a clique, need |Ni (vi )| ≤ ω(G) − 1 colors). Thus, at least one of the
ω(G) colors is still available for vi ; take such a color for vi . This also shows:
Corollary 3.3 Chordal graphs are perfect.
Actually, this also follows from the fact that weakly chordal graphs are perfect [267]
and chordal graphs are a subclass of them. While chordal graphs are a basic example
for graphs with tree structure, weakly chordal graphs and perfect graphs do not seem
to have any important tree structure.
Another example for the use of perfect elimination orderings of chordal graphs is a
linear time algorithm for Maximum Weight Independent Set on chordal graphs given
by András Frank [219]. Let G = (V, E) be a chordal graph with perfect elimination
ordering σ = (v1 , . . . , vn ) of G, and let S denote a maximum cardinality independent
set in G. For the unweighted case, it is clear that without loss of generality, v1 ∈ S
since S ∩ N[v1 ] = 1 (otherwise S would not be maximal) and if S ∩ N(v1 ) = {x} then
x can be replaced by v1 , i.e., S ′ := (S \ {x}) ∪ {v1 } is also a maximum cardinality
independent set in G. Thus the algorithm for Maximum Cardinality Independent Set
on chordal graphs goes along the p.e.o. σ from left to right, takes repeatedly a simplicial
vertex vi and deletes its neighbors in Gi = G[{vi , . . . , vn }].
For the weighted case the algorithm is a bit more complicated: Let G = (V, E) be a
chordal graph with perfect elimination ordering (v1 , . . . , vn ) of G and w : V −→ R+
a nonnegative weight function on V . The algorithm of Frank efficiently constructs a
maximum weight stable set I of G in the following way:
21
(0) I := ∅; all vertices in V are unmarked.
(1) for i := 1 to n do
if w(vi ) > 0 then mark vi and let w(u) := max(w(u) − w(vi ), 0) for all
vertices u ∈ Ni (vi ).
(2) for i := n downto 1 do
if vi is marked then let I := I ∪ {vi } and unmark all vertices u ∈ N(vi ).
Theorem 3.6 ([219]) The algorithm described above is correct and runs in linear
time.
Proof. It is clear that the algorithm runs in linear time. For the correctness, we need
the following (inductive) argument: As in the algorithm, let (v1 , . . . , vn ) be a p.e.o. of
G and w a weight function on V . Now let w ′ be the weight function resulting from
step (1) of the algorithm for the neighbors of the simplicial vertex v1 . We claim:
Claim 3.1.1 αw (G) = w(v1 ) + αw′ (G − v1 ).
This holds true by the following argument: Let S be a maximum-weight stable set
in G and let C = N(v1 ) denote the neighborhood of the simplicial vertex v1 ; C is a
clique. Clearly, |S ∩ C| ≤ 1. If for x ∈ C, w(v1 ) > w(x) holds then clearly x ∈
/ S otherwise w(S) could be enlarged by replacing x by v1 . Thus, in this case, line (1) of
the algorithm leads to w ′(x) = 0. If w(v1 ) = w(x) then if x ∈ S, x can also be replaced
by v1 .
Case 1. v1 ∈ S:
Then N(v1 ) ∩ S = ∅ and Claim 3.1.1 holds since αw (G) = w(v1 ) + αw (G[V \ N[v1 ]]) =
w(v1 ) + αw′ (G[V \ N[v1 ]]) = w(v1 ) + αw′ (G − v1 ) if S ∩ N(v1 ) = ∅.
Case 2. v1 ∈
/ S:
Then exactly one of the neighbors of v1 , say x, is in S (otherwise S would not be a
maximal stable set), and now w ′ (x) = w(x) − w(v1 ) > 0 holds and thus αw (G) =
w(x) + αw (G[V \ N[x]]) = w(v1 ) + (w(x) − w(v1 )) + αw (G[V \ N[x]]) = w(v1 ) + w ′(x) +
αw′ (G[V \ N[x]]) = w(v1 ) + αw′ (G − v1 ).
Then the marking part of the algorithm delivers a maximum weight independent set
in G.
2
It should be mentioned that various important NP-complete problems remain NPcomplete for (subclasses of) chordal graphs; examples are the Hamilton Cycle problem [370] (which remains NP-complete for strongly chordal split graphs) and the Efficient Domination problem [412]. The Minimum Dominating Set problem remains
NP-complete for split graphs [44].
22
3.1.3
Convexity in Chordal Graphs and Powers of Chordal Graphs
For graph G = (V, E), the k-th power Gk of G is defined as Gk = (V, E k ) with xy ∈ E k
if x 6= y and dG (x, y) ≤ k.
An induced subgraph H of G = (V, E) is an isometric subgraph of G if for all pairs of
vertices x, y ∈ V (H), their distance in H is the same as in G, i.e., dH (x, y) = dG (x, y).
A vertex set S ⊆ V is monophonically convex (m-convex for short) if for all pairs
x, y ∈ S and every induced path Pxy connecting x and y, each vertex of Pxy is also
contained in S.
Lemma 3.3 (Farber, Jamison [208]) If G is a chordal graph with perfect elimination ordering (v1 , . . . , vn ) of G then for all i, 1 ≤ i ≤ n, {vi , . . . , vn } is m-convex and
Gi = G[{vi , . . . , vn }] is isometric in G.
Chordal graphs are characterized by m-convexity in the following way:
Theorem 3.7 (Farber, Jamison [208]) The following are equivalent:
(i) G is chordal;
(ii) for all v ∈ V the disk D1 (v) = N[v] is m-convex;
(iii) for all v ∈ V and all j > 0 the disk Dj (v) with radius j is m-convex;
(iv) for all K ⊆ V which are m-convex N[K] is m-convex;
(v) for all K ⊆ V with K m-convex and all j > 0 the disks Dj (K) are m-convex.
While even powers of chordal graphs are in general not chordal (as the example of the
square of the 4-sun shows), odd powers of chordal graphs are known to be chordal. An
even stronger result is the following:
Theorem 3.8 (Duchet[198]) If Gk is chordal then Gk+2 is chordal.
LexBFS is helpful also in this case:
Theorem 3.9 ([68]) For a chordal graph G, every LexBFS ordering of G is a p.e.o.
of each odd power G2k+1 , k ≥ 1.
The proof of Theorem 3.9 in [68] is based on m-convexity properties of chordal graphs
as described above.
Corollary 3.4 ([68]) A graph G is chordal if and only if every LexBFS ordering of
G is a common p.e.o. of all odd powers of G.
23
This also implies that odd powers of a chordal graph are chordal.
In [58], it is shown:
Theorem 3.10 ([58]) There is a common p.e.o. for every chordal power of a chordal
graph.
Such a common p.e.o. is found by an algorithm called LMCS in [58] (which is a generalization of the Maximum Cardinality Search algorithm by Tarjan and Yannakakis
[421]); the time bound, however, is not linear but O(nm).
3.2
Split Graphs
Another subclass of chordal graphs which plays an important role in various contexts
is the following:
Definition 3.4 A graph is a split graph if its vertex set can be partitioned into a clique
and a stable set. Such a partition is called a split partition.
It is easy to see that the complement of a split graph is a split graph as well, and split
graphs are chordal.
Theorem 3.11 (Főldes, Hammer [214]) The following conditions are equivalent:
(i) G is a split graph.
(ii) G and G are chordal.
(iii) G is (2K2 , C4 , C5 )-free.
Proof. “(ii) ⇐⇒ (iii)”: If G and G are chordal then obviously G contains no induced
2K2 , C4 and C5 . In the other direction, note that for every k ≥ 6, Ck contains a 2K2 ,
and C5 = C5 . Thus, if G contains no induced 2K2 , C4 and C5 then G and G are
chordal.
“(i) =⇒ (ii)”: If the vertex set V of G has a partition into a clique Q and a stable set
S then obviously, every vertex in S is simplicial in G. Thus, a p.e.o. of G can start
with all vertices of S and finish with all vertices of Q. Similar arguments hold for G,
and thus G and G are chordal.
“(ii) =⇒ (i)”: Suppose that G and G are chordal (or, equivalently, G contains no
induced 2K2 , C4 and C5 ).
If there is a vertex v ∈ V which is simplicial in G and G then N[v] is a clique and N[v]
is a stable set giving the desired split partition.
If there is a vertex v ∈ V which is neither simplicial in G nor simplicial in G then
let a, b ∈ N(v) be vertices with ab ∈
/ E and let c, d ∈ N[v] with cd ∈ E. Since G is
24
2K2 -free, a sees c or d, and similarly, b sees c or d but since G is C4 -free, a and b do
not have a common neighbor in c, d. Thus, say, a sees c but not d and vice versa for b
but now v, a, b, c, d induce a C5 in G which is a contradiction.
Thus, every vertex v ∈ V is either simplicial in G or simplicial in G. Let V1 := {v ∈ V |
v is simplicial in G} and V2 := {v ∈ V | v is simplicial in G}. Note that V = V1 ∪ V2 is
a partition of V . Now, if V1 is a stable set and V2 is a clique then this gives the desired
split partition. Suppose to the contrary that V1 contains an edge xy ∈ E. Then since
G is 2K2 -free, the set of nonneighbors of x and y form a stable set, and since x and
y are simplicial, the set of neighbors of x and y form a clique which gives the desired
split partition.
2
Since chordality of G can be checked in linear time, using Theorem 3.11, one obtains
a linear time recognition of split graphs if one can check whether G is chordal without
explicitly constructing G; this has been done in [277].
The following nice characterization of split graphs in terms of their degree sequence
leads to an immediate linear time recognition of split graphs:
Theorem 3.12 (Hammer, Simeone 1981; Tyshkevich, Melnikov, Kotov 1981)
Let G have the degree sequence d1 ≥ d2 ≥ . . . ≥ dn and ω := max{i | di ≥ i − 1 for
i ∈ {1, . . . , n}}. Then G is a split graph if and only if
(∗) Σωi=1 di = ω(ω − 1) + Σni=ω+1 di
holds. Moreover, if (∗) holds then ω(G) = ω.
Proof. “=⇒”: Assume that G = (V, E) is a split graph with clique Q and stable set
S such that V = Q ∪ S and Q is a maximal clique in G. In the trivial case when S = ∅
and |Q| = ω, the degree sequence (ω −1, . . . , ω −1) obviously fulfills condition (∗) since
ω = n and Σni=n+1 di = 0. In the other trivial case when Q = ∅, the degree sequence
(0, 0, . . . , 0) obviously fulfills condition (∗).
Now assume that G = (V, E) is a nontrivial split graph with nonempty clique Q and
nonempty stable set S such that V = Q ∪ S and Q is a maximal clique in G. Then
for ω := |Q|, every vertex in Q has degree at least ω − 1 since Q is a clique, and every
vertex in S has degree at most ω−1 since S is a stable set. Thus, in the degree sequence
d1 ≥ d2 ≥ . . . ≥ dn , the first ω vertices are in Q, and Σωi=1 di − ω(ω − 1) = Σni=ω+1 di
since Σωi=1 di − ω(ω − 1) is the number of edges between Q and S which is the same as
the number of edges between S and Q, namely Σni=ω+1 di .
“⇐=”: Assume that (∗) holds, i.e., Σωi=1 di − ω(ω − 1) = Σni=ω+1 di . Then let A denote
the first ω vertices in the degree sequence d1 ≥ d2 ≥ . . . ≥ dn and B = V \ A.
Obviously, A 6= ∅. If B = ∅ (namely when the degree sequence is (ω − 1, . . . , ω − 1) for
ω = n) then G is a clique. Now assume that also B 6= ∅. We claim that A is a clique
and B is a stable set: Since |A| = ω, the degree sum according to the edges having
both vertices in A is at most ω(ω − 1). Let NA,B denote the number of edges between
A and B. Obviously, Σωi=1 di − ω(ω − 1) ≤ NA,B ≤ Σni=ω+1 di − 0, and thus, by (∗),
Σωi=1 di − ω(ω − 1) = NA,B = Σni=ω+1 di . This also means Σωi=1 di − NA,B = ω(ω − 1), i.e.,
25
Figure 2: Four examples of an AT, namely in C6 , 3-sun, net, and S2,2,2
the number of edges inside A is ω(ω − 1) and thus, A is a clique, and for B, we have
NA,B = Σni=ω+1 di − 0, i.e., Σni=ω+1 di − NA,B = 0 and thus, the number of edges inside
B is 0, i.e., B is a stable set.
2
Corollary 3.5 If G is a split graph and G′ has the same degree sequence then G′ is a
split graph as well.
3.3
Interval Graphs
In Theorem 3.2, chordal graphs are characterized as the intersection graphs of subtrees
of a tree. For interval graphs, the intersection model is much more special:
Definition 3.5 (Hajós [260]) A graph is an interval graph if it is the intersection
graph of finite closed intervals of the real line.
Based on the definition of chordal graphs as Ck -free graphs for every k ≥ 4, it is easy
to see that every interval graph is chordal, and a chordal graph with a clique tree being
a path is an interval graph.
Interval graphs are a very important subclass of chordal graphs. They are welldescribed in various monographs such as [236, 359, 396, 414]. They were characterized
in various ways as described below.
3.3.1
Characterizations of Interval Graphs
Three distinct vertices x, y, z form an asteroidal triple (AT for short) in a graph G
if x, y, z is an independent vertex set in G and for each two of them, there is a path
connecting these two but not containing any neighbor of the third. A graph is asteroidal
triple-free (AT-free for short) if it does not contain any asteroidal triple. In Figure 2,
the triples encapsulated in squares form AT’s.
Theorem 3.13 (Lekkerkerker, Boland [324]) A graph is an interval graph if and
only if it is AT-free and chordal.
26
Figure 3: Forbidden subgraphs of interval graphs
Proof. Obviously, every interval graph is AT-free and chordal. In [359], a proof of the
other direction of Theorem 3.13 is described using clique trees by showing that for an
AT-free chordal graph G, any clique tree of G with minimum number of leaves must
have exactly two leaves:
Let T be a clique tree of G with minimum number of leaves; if the number of leaves is 2
then clearly, G is an interval graph. Thus, assume that the number of leaves is at least
3; let Q1 , Q2 , Q3 be leaves of T , and let Q′i , i = 1, 2, 3, be the (uniquely determined)
neighbor of Qi in T . Obviously, there are vi ∈ Qi \ Q′i , i = 1, 2, 3. Since G is AT-free,
v1 , v2 , v3 is not an AT. Let T (Q1 , Q3 ) be the T -path connecting Q1 and Q3 . Since G
is AT-free, T (Q1 , Q3 ) must contain a T -edge Q′ Q′′ with Q′ ∩ Q′′ ⊆ NG (v2 ) such that
neither Q′ nor Q′′ are leaves in T . Without loss of generality, let Q′ be closer to Q2 in
T than Q′′ . Then the clique tree T can be replaced by a new clique tree T ′ by replacing
the T -edge Q′ Q′′ by Q2 Q′′ . Now T ′ has fewer leaves than T which is a contradiction.
2
Based on Theorem 3.13, a forbidden subgraph characterization of interval graphs is
given in [324] (see e.g. [359]):
Theorem 3.14 (Lekkerkerker, Boland [324]) A graph is an interval graph if and
only if it is chordal and contains none of the graphs in Figure 3 as an induced subgraph.
An undirected graph G = (V, E) is transitively orientable if there is an orientation
T ⊆ V × V of all edges such that T is transitive, i.e., if for edges xy ∈ E and yz ∈ E,
(x, y) ∈ T and (y, z) ∈ T implies (x, z) ∈ T .
Theorem 3.15 ([234]) G is an interval graph if and only if G is C4 -free and its
complement graph G is transitively orientable.
Interval graphs can also be characterized in terms of special clique trees:
Theorem 3.16 ([221]) A graph is an interval graph if and only if there is a clique
tree of its maximal cliques which is a path.
We collect these two results in the following:
Corollary 3.6 The following are equivalent:
27
(1) G is an interval graph.
(2) G is C4 -free and G is transitively orientable.
(3) The maximal cliques of G can be linearly ordered such that for every vertex v the
cliques containing v form an interval in this arrangement.
Proof. (1) =⇒ (2): First assume that G is an interval graph. Then, based on the
left-right partial order of intervals, G is a comparability graph and thus transitively
orientable. Assume that the vertices u, v, x, y induce a C4 in G with edges uv, vx, xy, yu.
Then Iu ∩ Ix = ∅ and Iv ∩ Iy = ∅. Without loss of generality, assume that Iu is on
the left of Ix . By Iu ∩ Iv 6= ∅ and Iv ∩ Ix 6= ∅, the nonempty interval Z between the
intervals Iu and Ix is contained in Iv : Z ⊆ Iv , and by symmetry, the same holds for Iy :
Z ⊆ Iy , thus Iv ∩ Iy 6= ∅ which is a contradiction.
(2) =⇒ (3): Now assume that G is C4 -free and its complement graph is transitively
orientable; let F be a transitive orientation of the edges in G. We are going to show
that the maximal anti-chains A1 , . . . , Ak in G fulfill the linear ordering property as
the clique path in Theorem 3.16. Note that A1 , . . . , Ak are the maximal cliques in G.
Between maximal anti-chains Ai and Aj , there is an edge x → y ∈ F since otherwise,
Ai ∪ Aj would be an anti-chain. Moreover, we have: If a → b and c → d are edges
between Ai and Aj then both edges either are directed from Ai to Aj or vice versa,
i.e., all edges between Ai and Aj have the same direction: Assume to the contrary that
a, d ∈ Ai , b, c ∈ Aj with (a, b), (c, d) ∈ F . Then at least one of the pairs bd, ac is in
E, otherwise G contains C4 (E contains 2K2 , respectively). Without loss of generality
let ac ∈ E. If a → c ∈ F then, by transitivity of F , also a → d ∈ F which is a
contradiction. If c → a ∈ F then also c → b ∈ F which is a contradiction. Thus, all
edges between Ai and Aj , i, j ∈ {1, . . . , k}, have the same direction. Thus, A1 , . . . , Ak
form a complete directed acyclic graph (a tournament) in the following sense: Ai → Aj
if and only if there is x ∈ Ai , y ∈ Aj with x → y ∈ F . We claim that if Ai → Aj and
Aj → Ak then also Ai → Ak : Let x → y ∈ F with x ∈ Ai , y ∈ Aj and u → v ∈ F
with u ∈ Aj , v ∈ Ak . Then for the case y = u, the claim obviously holds. If yv ∈ E
then y → v ∈ F and the claim holds, and similarly, if xu ∈ E then x → u ∈ F and the
claim holds. Now if yv ∈ E and xu ∈ E then xv ∈
/ E since otherwise G contains C4 .
Thus either x → v ∈ F or v → x ∈ F . If x → v ∈ F then Ai → Ak . It is impossible
that v → x ∈ F since in that case u → x ∈ F which is a contradiction.
Now assume that for some vertex x ∈ V x ∈ Ai and x ∈ Ak but x ∈
/ Aj holds. Since
x∈
/ Aj , there is a vertex y ∈ Aj with xy ∈ E. By Ai → Aj it follows that x → y ∈ F ,
and by Aj → Ak it follows that y → x ∈ F which is a contradiction.
(3) =⇒ (1): For every vertex x choose the interval Ix of cliques in G (i.e., anti-chains
in F ) containing x. This is an interval model M for G such that G = GM .
2
The following umbrella property characterizes interval graphs:
Theorem 3.17 (Olariu [383]) G = (V, E) is an interval graph if and only if there is
an ordering σ = (v1 , . . . , vn ) of V such that for all i < j < k holds: If vi vk ∈ E then
vi vj ∈ E.
28
Figure 4: net, S3 , rising sun, and co-rising sun
Recognition of interval graphs was done in linear time by Booth and Lueker [54] using
the famous PQ-trees. In [312] another linear time algorithm is given. Recently, LexBFS
was applied for a different approach by Corneil, Olariu and Stewart [152, 153] using
a 6-sweep LexBFS algorithm which improves an earlier attempt by K. Simon to use
LexBFS for efficiently recognizing interval graphs.
If G and its complement graph is an interval graph then G is a special kind of split
graphs namely a split graph and a permutation graph since G and G are chordal as well
as transitively orientable; see [93, 236, 414] for more details on permutation graphs.
These graphs have a nice characterization by forbidden induced subgraphs (see Figure
4):
Theorem 3.18 (Benzaken, Hammer, de Werra [31]) The following conditions
are equivalent:
(1) G and G are both interval graphs.
(2) G is a (S3 ,net,rising sun,co-rising sun)-free split graph.
(3) G is (2K2 , C4 , C5 , S3 ,net,rising sun,co-rising sun)-free
3.3.2
Unit Interval Graphs
An interval graph is a unit interval graph if it has an interval model with unit length
(closed) intervals. This concept appears already in [434]. An interval graph is a proper
interval graph if it has an interval model such that no two intervals contain each other.
G = (V, E) is an indifference graph if there are δ > 0 and f : V → R such that
xy ∈ E ⇐⇒ |f (x) − f (y)| ≤ δ. The following theorem mostly goes back to results of
Fred Roberts [394, 395]:
Theorem 3.19 The following conditions are equivalent:
(1) G is a unit interval graph.
(2) G is a proper interval graph.
(3) G is an indifference graph.
(4) G is a claw-free interval graph.
29
(5) There is an ordering σ = (v1 , . . . , vn ) of V such that for every vertex v ∈ V , the
vertices of its closed neighborhood are consecutive in σ.
(6) There is an ordering σ = (v1 , . . . , vn ) of V such that vertices contained in the
same maximal clique are consecutive in σ.
Roberts’ proof of the equivalence of unit and proper interval graphs is discussed in
[226]. The following umbrella property (similar to the previous one of Theorem 3.17)
characterizes unit interval graphs:
Theorem 3.20 ([339]) G = (V, E) is a unit interval graph if and only if there is an
ordering σ = (v1 , . . . , vn ) of V such that for all i < j < k holds: If vi vk ∈ E then
vi vj ∈ E and vj vk ∈ E.
A simple structural description of unit interval graphs (related to leaf powers) is given
by the following (see also [333] for this property):
Theorem 3.21 ([80, 81]) G is a unit interval graph if and only if G is an induced
subgraph of some k-th power Pnk of a chordless path Pn .
Proof. “ ⇐=′′ : Clearly, every chordless path Pn is a unit interval graph; an interval
model can be as follows: for vertex vi , 1 ≤ i ≤ n, interval Ii is the closed interval
[i, i + 1]. For the k-th power Pnk of Pn , we can take the intervals [i, i + k] as a unit
interval graph model. Clearly, every induced subgraph of such a graph is a unit interval
graph as well.
“ =⇒′′ : Let G = (V, E) be a unit interval graph. Without loss of generality, we can
assume that all intervals Iv , v ∈ V , are closed intervals Iv = [lv , rv ] with the same
integer length k and integers lv , rv . Then G is an induced subgraph of Pℓk for some
ℓ ≥ n.
2
k
This leads to a grid structure of the following type: Take for example Gk = Pk×k
:
the vertex set of Gk is Vk := {v1 , . . . , vk×k } and the set of edges is Ek := {vi vj | i 6=
j, |i − j| ≤ k}. Equivalently, Vk can be partitioned into k cliques Q1 , . . . , Qk such
that the edges between Qi and Qi+1 forms a chain graph (i.e., 2K2 -free - for this, Qi
and Qi+1 are assumed to be independent vertex sets) of the same type for every i,
1 ≤ i ≤ k − 1. In Figure 5, the 4-th power of P16 is shown.
Interestingly, the structure of bipartite permutation graphs is very similar as described
in [96]; the main difference is that for every i, Qi is not a clique but an independent
vertex set but the edges between Qi and Qi+1 forms a chain graph of the same type
for every i, 1 ≤ i ≤ k − 1.
Recognition of unit interval graphs can be done in linear time as described in [148,
181, 182, 272, 388]. In [146], a simple 3-sweep LexBFS algorithm is given. See [414]
for comments and a direct way for recognizing unit interval graphs in linear time.
30
4
Figure 5: P16
The situation changes if for unit interval models, open, half-open and closed intervals
are admitted (so-called mixed unit interval graphs). Claw, for instance, has the following intersection model with mixed unit intervals: Let a, b, c, d induce a claw with edges
ad, bd, cd, and let Ia = [0, 1], Ib = [2, 3], Ic = (1, 2), and Id = [1, 2].
In [295], a characterization of mixed unit interval graphs is given.
3.3.3
Generalizations and Variants of Interval Graphs and Related Graph
Classes
By Theorem 3.15 and the result of Ghouila-Houri [232] that a graph is a comparability graph if and only if it is transitively orientable, every interval graph is a cocomparability graph. By Dushnik and Miller’s result [200], a graph G is a permutation
graph if and only if G and G are comparability graphs. Co-comparability graphs are
AT-free (see [93]).
The linear structure of permutation graphs, co-comparability graphs, interval graphs
and their characterization by asteroidal triples and chordality - recall Theorem 3.13 - is
generalized by the larger class of AT-free graphs which still has a linear structure; many
papers deal with structure and algorithmic use of such graphs; for example, see the
seminal paper of Corneil, Olariu and Stewart [151] and the Ph.D. thesis of Ekkehard
Köhler [310]. Moreover, the following holds (a minimal triangulation of G = (V, E)
adds a minimal set of edges to E, thus obtaining E ′ with E ⊆ E ′ such that G′ = (V, E ′ )
is chordal):
Theorem 3.22 ([363, 151, 389]) G is an AT-free graph if and only if every minimal
triangulation of G is an interval graph.
As an example for the algorithmic use of AT-free graphs, in [310], an algorithm with
time bound O(n3 ) is given for solving the Maximum Weight Independent Set problem
on AT-free graphs. See [414] for various other examples. Important subclasses of ATfree graphs are the permutation graphs and the larger class of co-comparability graphs
[243]. The class of bipartite AT-free graphs is the same as bipartite permutation
graphs as can be shown by using a forbidden induced subgraph characterization of
comparability graphs found by Gallai [223] (see [93, 414]). This also follows from
the characterization of AT-free graphs by forbidden induced subgraphs [310] (which is
based on Gallai’s result).
31
An interesting bipartite variant of interval graphs called interval bigraphs (in [359], it
is called bipartite interval graphs) is the following:
Definition 3.6 (Harary, Kabell, McMorris [266]) A bipartite graph B
=
(X, Y, E) is an interval bigraph if there are families of intervals IX = {I1 , . . . , Ir },
r = |X|, and IY = {J1 , . . . , Js }, s = |Y |, such that xi yj ∈ E if and only if Ii ∩ Jj 6= ∅
for i ∈ {1, . . . , r}, j ∈ {1, . . . , s}.
Müller [369] gives a polynomial time recognition of interval bigraphs. Moreover he
shows that convex graphs are interval bigraphs, and interval bigraphs are chordal bipartite. The recognition time bound is substantially improved in [390] to O(nm). If
one generalizes intervals Ix , Iy to subtrees of a tree then every bipartite graph can be
represented by such an intersection model [266].
Another extremely important generalization of interval graphs is given by tolerance
graphs introduced in [243]; see the monograph [245] of Golumbic and Trenk for a
collection of results on these graphs. A graph G = (V, E) is a tolerance graph if there
is a collection I = {Iv | v ∈ V } of intervals on the real line and a set T = {tv | v ∈ V }
of positive numbers (the tolerances) such that for any two distinct vertices u, v ∈ V ,
uv ∈ E if and only if |Iu ∩ Iv | ≥ min{tu , tv } holds (|I| denotes the length of the interval
I). In this case, the pair (I, T ) is a tolerance representation of G. If for all v ∈ V ,
tv ≤ |Iv | holds then (I, T ) is a bounded tolerance representation of G and G is a bounded
tolerance graph. Tolerance graphs are weakly chordal [243]. Solving a long-standing
open problem, it was shown by Mertzios, Sau and Zaks in [360] that the recognition of
tolerance graphs and of bounded tolerance graphs is NP-complete.
The class of bipartite bounded tolerance graphs is the same as bipartite permutation
graphs [414]. In Theorem 3.9 of [245], the equivalence of the various classes is collected
as follows:
Theorem 3.23 Let B = (X, Y, E) be a bipartite graph. The following conditions are
equivalent:
(i) B is a bounded tolerance graph.
(ii) B is a trapezoid graph.
(iii) B is a cocomparability graph.
(iv) B is AT-free.
(v) B is a permutation graph.
(vi) B has the SBS-indexing property.
(vii) There are ε > 0 and a real-valued function f on X ∪ Y such that xy ∈ E ⇐⇒
|f (x) − f (y)| ≤ ε for x ∈ X and y ∈ Y .
32
Figure 6: F1 ,F2 , and F3
Graphs fulfilling Condition (vii) were called bipartite tolerance graphs in [104]. The
class of bipartite tolerance graphs in the sense of [243] properly contains bipartite
bounded tolerance graphs; bipartite tolerance graphs are characterized in [115]; the
characterization implies that these graphs form a proper subclass of interval bigraphs.
In [310], another characterization of AT-free bipartite graphs is given in terms of forbidden induced subgraphs (see Figure 6).
Theorem 3.24 G is a bipartite AT-free graph if and only if G is C2k -free for k ≥ 3
and F1 , F2 , F3 -free.
Another important generalization of interval graphs is given by circular-arc graphs: G
is a circular-arc graph if it is the intersection graph of arcs of a circle. This generalizes
the linear structure of interval graphs to circular structure ; see e.g. [414] for many details and McConnell’s paper [354] giving linear time recognition of circular-arc graphs.
Clearly, circular-arc graphs are not perfect since e.g. C5 is a circular-arc graph.
3.4
Chords and Cycles
Chordal graphs can be generalized in a natural way by placing a variety of restrictions
on the number and type of chords with respect to a cycle. A fairly general scheme is
given in the following definition (which was motivated by relational database schemes):
Definition 3.7 (Ausiello, D’Atri, Moscarini [11]) For k ≥ 4 and ℓ ≥ 1, a graph
G is (k, ℓ)-chordal if each cycle in G of length at least k contains at least ℓ chords.
Thus chordal graphs are the (4,1)-chordal graphs. Further conditions can be placed on
the parity of the cycles (chords in odd cycles), the parity of the cycle distance of the
end vertices of chords (odd chords), requiring crossing and/or parallel chords, requiring
all these conditions for G and G, and requiring these conditions in bipartite graphs
(where all cycles are of even length). Thus, for example, the (5,2)-odd-crossing-chordal
graphs are the graphs such that every odd cycle of length at least five has at least two
crossing chords.
See [93] for more details and Theorem 8.5 for a characterization of (5,2)-chordal graphs.
33
Finally, another interesting subclass of chordal graphs should be mentioned which will
be discussed in more detail in the section on strongly chordal graphs and on β-acyclicity.
Assume that G is a chordal graph. A chord xi xj in a cycle C = (x1 , x2 , . . . , x2k , x1 ) of
even length 2k is an odd chord if the distance in C between xi and xj is odd.
Farber [207] defined strongly chordal graphs in terms of strong elimination orderings
rather than odd chords in even cycles (see Definition 7.2), but he showed that chordal
graphs having odd chords in even cycles are exactly the strongly chordal graphs (see
Theorem 7.6).
34
4
4.1
α-Acyclic Hypergraphs and Their Duals
A Motivation from Relational Database Theory
Fagin [204] gives a very nice introduction into acyclic database schemes (of various
degrees, namely α-, β- and γ-acyclicity) and their equivalence to desirable properties
of relational databases. Since Fagin’s introduction is mostly informal and we need some
definitions, we follow the presentation in papers such as [28] for this subsection.
A (relational) database scheme as introduced by Codd [144] can be thought of as a
collection of table skeletons, or, alternatively, as a set of subsets of attributes, or column
names in the tables. These attribute subsets form the hyperedges of a finite hypergraph.
A relational database corresponds to a family of relations over the attributes.
Let V = {v1 , . . . , vn } be a finite set of distinct symbols called attributes or column names
(for example, name, first name, age, birthday, citizenship, married, home address,
telephone number etc).
Let Y ⊆ V . A Y -tuple is a mapping that associates a value (from a certain universe
U) with each attribute in Y . For instance, if Y = {name, age, citizenship, married}
then a Y -tuple is a 4-tuple such as (Higgins, 48, Canada, no).
If X ⊆ Y and t is a Y -tuple, then the projection t[X] denotes the X-tuple obtained by restricting t to X. For instance, if X = {name, citizenship} and t =
(Higgins, 48, Canada, no) then t[X] = (Higgins, Canada).
A Y -relation is a finite set of Y -tuples. If r is a Y -relation and X ⊆ Y then by the
projection r[X] of r onto X, we mean the set of all tuples t[X], where t ∈ r.
If V is a set of attributes, then we define a relational database scheme (database scheme
for short) E = {E1 , . . . , Em } to be a set of subsets of V , i.e., (V, E) is a hypergraph
over vertex set V .
Intuitively, for each i, the set Ei of attributes is considered to be the set of column
names for a relation; the Ei ’s are called relation schemes. If r1 , . . . , rm are relations,
where ri is a relation over Ei , i ∈ {1, . . . , m}, then we call {r1 , . . . , rm } a database over
E.
The join r1 ⊲⊳ r2 of two relations r1 and r2 with attribute sets E1 and E2 , respectively,
is the set of all tuples t with attribute set E1 ∪ E2 for which the projection t[Ei ] is in
ri , i = 1, 2.
Example 4.1
r1 : A B
0 0
1 1
r1 ⊲⊳ r2 : A B C
0 0 0
1 1 1
r2 : B C
0 0
1 1
r3 : A C
0 1
1 0
r1 ⊲⊳ r3 : A B C r2 ⊲⊳ r3 : A B C
0 0 1
1 0 0
1 1 0
0 1 1
35
More generally, the join r1 ⊲⊳ . . . ⊲⊳ rm of the relations r1 , . . . , rm , m ≥ 2, with attribute
sets E1 , . . . , Em , respectively, is the set of all tuples t with attribute set E1 ∪ . . . ∪ Em ,
such that for each i ∈ {1, . . . , m}, the projection t[Ei ] of tuple t onto attributes Ei
fulfills t[Ei ] ∈ ri .
The join of all three relations in Example 4.1 is empty: r1 ⊲⊳ r2 ⊲⊳ r3 = ∅.
We say that a relation r with attributes E1 ∪ . . . ∪ Em obeys the join dependency
⊲⊳ {E1 , . . . , Em } if r = r1 ⊲⊳ . . . ⊲⊳ rm , where ri = r[Ei ] for each i ∈ {1, . . . , m}.
A highly desirable property of a relational database r1 , . . . , rm , m ≥ 2, is that the
entries in it are conflict-free. In general, the attribute sets are not pairwise disjoint,
and it easily might happen that an entry in one of the relations is updated while the
same entry in another relation is not. Pairwise consistency captures conflict-freeness
for every two of the relations, and global consistency, roughly saying, means that all
of them together are conflict-free. If the relations are globally consistent then they are
pairwise consistent but not vice versa as Example 4.1 shows; surprisingly, it turns out
that the equivalence of pairwise and of global consistency corresponds to a hypergraph
acyclicity property of the underlying attribute sets.
More formally, let r and s be relations with attributes R and S, respectively, and let
Q = R ∩ S, i.e., Q is precisely the set of attributes that r and s have in common. We
say that r and s are consistent if r[Q] = s[Q], i.e., the projections of r and s onto their
common attributes are the same.
Example 4.2
r1 : A
0
1
2
B
1
2
3
C
2
3
4
r2 : A
0
0
3
D
3
5
4
E
4
6
5
r1 ⊲⊳ r2 : A B C D E
0 1 2 3 4
0 1 2 5 6
In Example 4.2, r1 and r2 have only A as common attribute, and the projection r1 [A]
is {0, 1, 2} while the projection r2 [A] is {0, 3}; thus, r1 and r2 are not consistent.
In Example 4.1, each pair ri , rj of relations, i, j ∈ {1, 2, 3}, is consistent.
Definition 4.1 Let {r1 , . . . , rm } be an arbitrary database over E = {E1 , . . . , Em }.
(i) {r1 , . . . , rm } is pairwise consistent if for all i, j ∈ {1, . . . , m}, ri and rj are consistent.
(ii) {r1 , . . . , rm } is globally consistent if there is a relation r over the attribute set
E1 ∪ . . . ∪ Em such that for each i ∈ {1, . . . , m}, ri = r[Ei ]. Then r is called
universal for {r1 , . . . , rm }.
36
Thus, {r1 , . . . , rm } is globally consistent if and only if there is a (universal) relation r
such that each ri is the projection of r onto the corresponding attribute set of ri . Such
a universal relation need not be unique, but it is known that if there is such a universal
relation r, then also r1 ⊲⊳ . . . ⊲⊳ rm is such a universal relation:
Lemma 4.1 If r is a universal relation for r1 , . . . , rm with attribute sets E1 , . . . , Em
then r ⊆ r[E1 ] ⊲⊳ . . . ⊲⊳ r[Em ] = r1 ⊲⊳ . . . ⊲⊳ rm .
It is clear that if {r1 , . . . , rm } is globally consistent then it is pairwise consistent but
in general, the converse is false as the relations r1 , r2 , r3 in Example 4.1 show which
are pairwise consistent but not globally consistent.
Honeyman, Ladner and Yannakakis [280] have shown:
Theorem 4.1 ([280]) The global consistency of a relational database is an NPcomplete problem.
In [28], it is shown that for a relational database scheme, pairwise consistency implies
global consistency if and only if it is α-acyclic (see Theorem 4.8).
4.2
4.2.1
Hypergraph Parameters: Coloring, Packing and Covering
Hypergraph 2-Coloring
A hypergraph H = (V, E) is 2-colorable if its vertex set V has a partition V = V1 ∪ V2
such that every hyperedge e ∈ E has at least one vertex from each of the sets V1 and
V2 . See [33] for the more general notion of hypergraph coloring. The Hypergraph 2Coloring Problem (also called Bicoloring Problem, Set Splitting Problem [SP4] in [224])
is the question whether a given hypergraph is 2-colorable.
Lovász [340] has shown that the Hypergraph 2-Coloring Problem is NP-complete even
for hypergraphs whose hyperedges have size at most 3 (see [224]); the original reduction
in [340] is from the Graph Coloring Problem (which has been shown to be NP-complete
in [298]) to Hypergraph 2-Coloring.
The following nice reduction from the Satisfiability Problem SAT to the Hypergraph
2-Coloring Problem was given in [335].
Let F = C1 ∧ . . . ∧ Cm be a Boolean expression in conjunctive normal form (CNF
for short) with clauses C1 , . . . , Cm and variables x1 , . . . , xn . Each clause consists of a
disjunction of literals, i.e., unnegated or negated variables.
Let HF = (VF , EF ) be the following hypergraph for F :
The vertex set VF = {x1 , . . . , xn }∪{¬x1 , . . . , ¬xn }∪{f } where f is a new symbol
different from the variable symbols.
37
The edge set EF of HF consists of the following edges:
(i) For all i ∈ {1, . . . , n}, let Xi = {xi , ¬xi }
(ii) For all j ∈ {1, . . . , m}, let Yj be the set of all literals in Cj plus, additionally,
the element f .
We show that F is satisfiable if and only if HF is 2-colorable:
Given a truth assignment which satisfies F , we associate with it the following 2-coloring
V1 ∪ V2 : If xi has truth value 1 then xi ∈ V1 and ¬xi ∈ V2 and vice versa if xi has truth
value 0. The element f belongs to V2 . Now, for each i ∈ {1, . . . , n}, the edge {xi , ¬xi }
intersects both V1 and V2 . An edge Yi intersects V2 on f and intersects V1 since it has
a true literal.
On the other hand, given a 2-coloring V1 ∪ V2 of HF , with, say f ∈ V2 we assign true
to each xi in V1 and false to those in V2 . This gives a truth assignment since the edges
{xi , ¬xi } meet both V1 and V2 . The edge Yj of every clause Cj meets V1 on an element
other than f which ensures that every clause is satisfied. This shows
Theorem 4.2 ([340]) The 2-coloring problem for hypergraphs is NP-complete.
Based on Theorem 4.2, in [340], Lovász has shown:
Theorem 4.3 ([340]) The 3-coloring problem for graphs is NP-complete.
See [225] for another proof of the NP-completeness of the 3-coloring problem.
4.2.2
Matchings and Transversals: The Kőnig Property
The following definition generalizes the fundamental notions of matching and vertex
cover in graphs to the corresponding notions in hypergraphs.
Definition 4.2 Let H = (V, E) be a hypergraph.
(i) An edge set E ′ ⊆ E is called matching (or packing) if the edges of E ′ are pairwise
disjoint. The matching number (or packing number ) ν(H) of H is the maximum
number of pairwise disjoint hyperedges of H.
(ii) A transversal of E is a subset U ⊆ V such that U contains at least one vertex of
every e ∈ E. The transversal number τ (H) is the minimum number of vertices
in a transversal of H.
(iii) H has the Kőnig Property if ν(H) = τ (H).
38
Obviously, a packing E ′ in H = (V, E) corresponds to an independent set in L(H).
Note that for every hypergraph, ν(H) ≤ τ (H) holds. A well-known theorem of Kőnig
states that for bipartite graphs G, ν(G) = τ (G) holds. This justifies the name Kőnig
property and is closely related to the celebrated max-flow min-cut theorem by Ford and
Fulkerson.
Karp’s famous list of 21 NP-complete problems [224, 298] contains various problems
which are related to hypergraphs; here we mention three examples of them:
1. For a given hypergraph H and a natural number k, the Set Packing Problem asks
whether H has a packing of at least k edges.
2. The 3-Dimensional Matching Problem is a special case of the Set Packing Problem. +++++++++++++++++
3. For a given hypergraph H = (V, E), the Exact Cover Problem asks whether there
is a subset E ′ ⊆ E such that E ′ forms a partition of V , i.e., every vertex v ∈ V is in
exactly one hyperedge from E ′ . The Exact Cover Problem remains NP-complete
for hyperedges of size 3; for hyperedges of size 2, it corresponds to the Perfect
Matching Problem for graphs.
4.3
Tree Structure of α-Acyclic Hypergraphs
Unlike the case of graphs, there is a bewildering diversity of cycle notions in hypergraphs, and some of them play an important role in connection with desirable properties
of relational database schemes [28, 203, 204, 206, 246]. Thus, for example, the desirable
property of a relational database scheme that pairwise consistency should imply global
consistency turns out to be equivalent to α-acyclicity of the scheme [28, 203]; as shown
in Theorems 4.7 and 4.8, a relational database scheme has this property if and only if
it is α-acyclic. Moreover, α-acyclicity is equivalent to many other desirable properties
of such schemes. It seems that for applications in databases and other fields, the most
important property of an α-acyclic hypergraph is to have a join tree:
Definition 4.3 Let H = (V, E) be a hypergraph.
(i) Tree T is a join tree of H if the node set of T is the set E of hyperedges and for
every vertex v ∈ V , the set Ev of hyperedges containing v forms a subtree in T .
(ii) H is α-acyclic if H has a join tree.
Note that in this way, α-acyclicity of a hypergraph is defined without referring to any
cycle notion in hypergraphs.
Tarjan and Yannakakis [421] gave a linear-time algorithm for testing α-acyclicity of a
given hypergraph.
39
Tree structure in hypergraphs has been captured in the hypergraph community as
arboreal hypergraphs [33] (as well as its dual version, the co-arboreal hypergraphs) and
tree-hypergraphs in [321]. We call arboreal hypergraphs hypertrees:
Definition 4.4 A hypergraph H = (V, E) is a hypertree if there is a tree T whose set
of nodes is V and such that every hyperedge e ∈ E induces a subtree in T .
Note that in [247], Gottlob, Leone and Scarcello define the notion of hypertrees in a
completely different way.
The following properties are easy to see:
Proposition 4.1 Let H = (V, E) be a hypergraph.
(i) H is a hypertree if and only if its dual H ∗ is α-acyclic.
(ii) If H is a hypertree then every edge-subhypergraph of H is a hypertree as well but
not necessarily every vertex-subhypergraph of H.
(iii) If H is α-acyclic then every vertex-subhypergraph of H is α-acyclic as well but
not necessarily every edge-subhypergraph of H.
The fact that α-acyclic hypergraphs may contain hyperedge cycles of a certain
kind (there are various cycle definitions in hypergraphs), and the fact that edgesubhypergraphs of α-acyclic hypergraphs are not necessarily α-acyclic are somewhat
counterintuitive in comparison with cycles in graphs and led Goodman and Shmueli
[246] to the name tree schema for α-acyclic hypergraphs (see also [237] for a discussion).
The following theorem gives an important characterization of hypertrees (α-acyclic
hypergraphs, respectively).
Theorem 4.4 (Duchet [197], Flament [213], Slater [411]) A hypergraph H is a
hypertree if and only if H has the Helly property and its line graph L(H) is chordal.
Proof. “=⇒”: Let H = (V, E) be a hypertree and let T be a tree with vertex set V
such that for all e ∈ E, T [e] induces a subtree in T . By Corollary 2.3, H has the Helly
property. By Proposition 3.1, L(H) is chordal.
“⇐=”: A dual variant of the assertion is the following: If H is conformal and 2SEC(H)
is chordal then H is α-acyclic. Without loss of generality we may assume that no
hyperedge of H is contained in another one. By Lemma 2.1, H is the clique hypergraph
of its 2-section graph, and by Theorem 3.4, the chordal graph 2SEC(H) has a clique
tree. Thus, H is α-acyclic.
2
By Proposition 2.2 and 2.3, Theorem 4.4 can also be formulated in the following equivalent way:
40
Corollary 4.1 H is α-acyclic if and only if H is conformal and 2SEC(H) is chordal.
See Definition 4.2 for the Kőnig property. As a consequence of Theorem 4.4, we obtain:
Corollary 4.2 Hypertrees have the Kőnig property.
Proof. Let H = (V, E) be a hypertree. Then by Theorem 4.4, H has the Helly
property and since L(H) is chordal, there is a p.e.o. (e1 , . . . , em ) of the set E of
hyperedges. Since e1 is simplicial in L(H), the set E1 of hyperedges intersecting e1 is
pairwise intersecting. By the Helly property, there is a vertex v in the intersection
of E1 . Now assume inductively that the hypergraph H ′ = (V, E \ E1 ) fulfills already
the condition τ (H ′ ) = ν(H ′ ). A maximum packing of H consists of a packing of H ′
and one additional hyperedge from E1 , and a minimum transversal of H consists of a
minimum transversal of H ′ and additionally the vertex v. Thus, also τ (H) = ν(H)
holds.
2
4.4
Join Trees as Maximum Spanning Trees
The α-acyclicity of a hypergraph H can also be characterized in terms of an inequality
concerning the weighted line graph of H. This was shown by Acharya and Las Vergnas
in the hypergraph community (see Theorem 4.5) but was also discovered by Bernstein
and Goodman [36] in the database community.
Definition 4.5 Let H = (V, E) be a hypergraph with E = {e1 , . . . , em }.
Let Lw (H) denote the weighted line graph of H whose nodes are the hyperedges of
H which are pairwise adjacent in Lw (H), and the edges are weighted by w(ei ej ) =
|ei ∩ ej | for i 6= j.
For any edge set F of L(H), let w(F ) denote the sum of all edge weights in F .
Let wH denote the maximum weight of a spanning tree in Lw (H).
For a spanning tree T of Lw (H), let Tv denote the subgraph of T induced by the
hyperedges Ev containing v, let N(Tv ) denote its node set and E(Tv ) its edge set.
If T is a spanning tree of Lw (H) then obviously for every vertex v ∈ V , the following
inequality holds:
1 ≤ |N(Tv )| − |E(Tv )|.
(2)
Since any tree with k ≥ 2 nodes has k − 1 edges, equality holds in (2) if and only if
Tv is a subtree of T . The following lemma summarizes what is implicitly contained in
Theorem 4.5.
41
e2
c
2
1
d
b
1
2
a
f
e
e1
e4
2
1
e3
Figure 7: A 3-sun with its four maximal cliques e1 = {a, b, f }, e2 = {b, c, d}, e3 =
{d, e, f } and e4 = {b, d, f } and the weighted line graph of them.
Lemma 4.2 (McKee [356]) Let H = (V, E) be a hypergraph with E = {e1 , . . . , em }
and let Lw (H) be as in Definition 4.5. Then a spanning tree T of Lw (H) is a join tree
of H if and only if
|V | = Σm
(3)
j=1 |ej | − Σij∈E(T ) |ei ∩ ej |.
Proof. Suppose T is a spanning tree of Lw (H). For each v ∈ V , the subgraph Tv
consisting of all hyperedges containing v satisfies 1 ≤ |N(Tv )| − |E(Tv )| as described
in (2), with equality if and only if, for all v ∈ V , Tv is connected. Summing over all
v ∈ V in (2) proves that inequality
|V | ≤ Σm
j=1 |ej | − Σij∈E(T ) |ei ∩ ej |
(4)
holds, and equality holds in (4) if and only if the spanning tree T is a join tree of H.
2
Note that the result of summing the right hand side of (2) is Σm
j=1 |ej | − w(T ) for the
spanning tree T of Lw (H). Thus also
m
|V | ≤ Σm
j=1 |ej | − max{w(T ) | T spanning tree of Lw (H)} = Σj=1 |ej | − wH
(5)
with equality in (5) if and only if H has a join tree.
Inequality (5) led to the following parameter (see Acharya and Las Vergnas [3], Hansen
and Las Vergnas [264], Lewin [325]):
Definition 4.6 Let H = (V, E) be a hypergraph and wH as in Definition 4.5. The
cyclomatic number µ(H) of H is defined as
µ(H) = Σm
j=1 |ej | − |V | − wH .
Note that the cyclomatic number of a hypergraph can be efficiently determined by any
maximum spanning tree algorithm such as the one by Kruskal [316]; see e.g. [145].
Now, the following theorem is a simple corollary of Lemma 4.2:
Theorem 4.5 (Acharya, Las Vergnas [3]) A hypergraph H satisfies µ(H) = 0 if
and only if H is α-acyclic.
42
Note that Lemma 4.2 respectively Theorem 4.5 suggests a way how to find a join tree of
an α-acyclic hypergraph, namely, taking any maximum spanning tree of the weighted
line graph Lw (H). Independently, this has been discovered in the database community
by Bernstein and Goodman [36] and rediscovered several times; see Chapter 2 of the
monograph [359] by McKee and McMorris.
However, this is not the most efficient way to construct a clique tree of a given chordal
graph; Theorem 3.5 gives a linear time algorithm for constructing a clique tree.
4.5
Graham’s Algorithm, Running Intersection Property and
Other Desirable Properties Equivalent to α-Acyclicity
In this subsection, we collect some properties which are equivalent to α-acyclicity of
a hypergraph. Some of these conditions are desirable properties of relational database
schemes as mentioned in the introduction. Beeri et al. [28] give a long list of such
equivalences; we mention here only some of them and give a few proofs which might
be suitable for a first glance at this field of research.
In Corollary 3.1 we have seen: A graph G is chordal if and only if G has a p.e.o.
A generalization of this for α-acyclic hypergraphs is known under the name Graham’s
Algorithm (or Graham Reduction):
Definition 4.7 (Graham [248], Yu, Ozsoyoglu [441]) Let H = (V, E) be a hypergraph.
(i) Graham’s Algorithm on H applies the following two operations to H repeatedly
as long as possible:
(1) If a vertex v ∈ V is contained in exactly one hyperedge e ∈ E then delete v
from e.
(2) If a hyperedge e is contained in another hyperedge e′ then delete e.
(ii) Graham’s Algorithm succeeds on H if repeatedly applying the two operations leads
to E = {∅}.
Graham’s algorithm is also called GYO algorithm since Yu and Ozsoyoglu [441] came
to exactly the same algorithm. Vertices which occur in only one edge are frequently
called ear vertices (isolated vertices in [28]) and edges containing such a vertex are
frequently called ears (knobs in [28]). Note that any ear node in H is simplicial in the
2-section graph of H.
Theorem 4.6 ( Beeri et al. [28], Goodman, Shmueli [246]) H is α-acyclic if
and only if Graham’s Algorithm succeeds on H.
43
Proof. “=⇒”: Let H be α-acyclic, i.e., by Corollary 4.1, H is conformal and 2SEC(H)
is chordal. If H is not Sperner then the (possibly repeated) application of rule (2) leads
to a Sperner hypergraph H ′ which is conformal and for which 2SEC(H ′) is chordal.
By Lemma 2.1, H ′ is isomorphic to the maximal clique hypergraph C(2SEC(H ′)).
Let (v1 , . . . , vn ) be a p.e.o. of 2SEC(H ′ ). Then v1 is simplicial and thus contained in
only one hyperedge of H ′ , i.e., v1 can be deleted by rule (1). Now the same argument
can be repeated and shows the assertion.
“⇐=”: Assume that Graham’s Algorithm succeeds on H. Then, the repeated application of rules (1) and (2) defines a vertex ordering σ = (v1 , . . . , vn ) of V (i.e., the
ordering in which by rule (1), the vertices get deleted). We claim that σ is a p.e.o. of
2SEC(H): Indeed, for each i ∈ {1, . . . , n}, when (1) is applicable to vi , this vertex
is contained in only one hyperedge and thus is simplicial in the remaining 2-section
graph.
We finally show that H is conformal: Let C be a clique in 2SEC(H) and let vi be its
leftmost element in σ. Then, when eliminating this vertex by rule (1), vi is contained in
only one hyperedge, say e, and all its neighbors in the 2-section graph are in e including
C, i.e., C ⊆ e which means that H is conformal.
2
Note that Graham’s algorithm produces a perfect elimination ordering of the 2-section
graph of H if H is α-acyclic.
The Running Intersection Property is another notion from the database community
which turns out to be equivalent to α-acyclicity of a hypergraph:
Definition 4.8 ([28]) Let H = (V, E) be a hypergraph. H has the running intersection
property if there is an ordering (e1 , e2 , . . . , em ) of E such that for all i ∈ {2, . . . , m},
there is a j < i such that ei ∩ (e1 ∪ . . . ∪ ei−1 ) ⊆ ej .
Theorem 4.7 (Beeri et al. [28], Goodman, Shmueli [246]) A hypergraph is αacyclic if and only if it has the running intersection property.
Proof. “=⇒”: If the hypergraph H = (V, E) is α-acyclic then it has a join tree T .
Select a root for T . Let (e1 , . . . , em ) be an ordering of E by increasing depth. Thus, if
ej is the parent of ei , then j < i. Clearly, each path from ei to any of e1 , . . . , ei−1 must
pass through ei ’s parent ej . Now if v ∈ V is a vertex in ei ∩ ek for some k < i, then
all hyperedges along the T -path between ei and ek contain v. Since this path passes
through ej , it follows that v ∈ ej which implies ei ∩ (e1 ∪ . . . ∪ ei−1 ) ⊆ ej . Thus, H has
the running intersection property.
“⇐=”: Let H = (V, E) be a hypergraph and let (e1 , . . . , em ) be an ordering of E fulfilling
the running intersection property. The proof is by induction on the number m of
hyperedges. The basis m = 2 is trivial. (e1 , . . . , em−1 ) also has the running intersection
property, and by induction hypothesis, there is a join tree T ′ for e1 , . . . , em−1 . Let T
be obtained from T ′ by adding node em and edge em ej for a j such that em ∩ (e1 ∪ . . . ∪
em−1 ) ⊆ ej . Obviously, T is a join tree for e1 , . . . , em .
2
For the next theorem, we need a few more definitions.
44
A path between two vertices u, v ∈ V in hypergraph H = (V, E) is a sequence of
k ≥ 1 edges e1 , . . . , ek ∈ E such that u ∈ e1 , v ∈ ek and for all i = 1, . . . , k − 1,
ei ∩ ei+1 6= ∅.
H is connected if for all pairs u, v ∈ V , there is a path between u and v in H.
The connected components of H
subhypergraphs of H.
are the maximal connected vertex-
For a reduced hypergraph H = (V, E) and two edges e, e′ ∈ E, e ∩ e′ is an edgeintersection-separator, e.i.-separator for short (called an articulation set in [28]) if
the reduced vertex-subhypergraph H[V \(e∩e′ )] has more connected components
than H.
A hypergraph H = (V, E) is edge-intersection-separable, e.i.-separable for short
(called acyclic in [28]) if for each U ⊆ V , if the reduction of H[U] is connected
and has more than one edge (i.e., is nontrivial) then it has an edge-intersectionseparator.
A hyperedge subset
F ⊆ E is closed if for each e ∈ E, there is an edge f ∈ F
S
such that e ∩ F ⊆ f .
A reduced hypergraph H = (V, E) is closed-e.i.-separable (called closed-acyclic
in [28]) if for each U ⊆ V , if H[U] is connected and has more than one edge
and its set of edges is closed then it has an e.-i.-separator. A hypergraph is
closed-e.i.-separable if its reduction is.
Note that in this definition, separators are always intersections of edges.
In [205], it is shown that a hypergraph is acyclic if and only if it is closed-acyclic, that is,
e.i.-separable and closed-e.i.-separable are equivalent notions. This has the advantage
that it is not necessary to deal with partial edges that are not edges.
Recall that in section 4.1, pairwise and global consistency, semijoins and full reducers,
monotone join expressions, and monotone sequential join expressions are defined.
Apparently, there is a close connection between the Helly property of a hypergraph and
the equivalence between pairwise and global consistency of a relational database scheme
(see Definition 4.1): A relational database r1 , . . . , rm over scheme E = {e1 , . . . , em } is
pairwise consistent if for every pair i, j ∈ {1, . . . , m}, ri , rj is consistent. Let Ri ,
i ∈ {1, . . . , m}, denote the set of relations over at least the attributes ei such that
the projection to ei is ri . In other words, pairwise consistency means that for all
i, j ∈ {1, . . . , m},Tthe intersection Ri ∩ Rj is nonempty. Global consistency means that
the intersection m
i=1 Ri is nonempty.
The next theorem is part of the main theorem in [28] which contains various other
conditions. See the same paper for a detailed discussion of other papers where parts
of these equivalences were shown.
45
Theorem 4.8 (Beeri et al. [28]) Let E be a hypergraph. The following conditions
are equivalent:
(i) E has the running intersection property.
(ii) E has a monotone sequential join expression.
(iii) E has a monotone join expression.
(iv) every pairwise consistent database over E is globally consistent.
(v) E is closed-e.i.-separable.
(vi) every database over E has a full reducer.
(vii) the GYO reduction algorithm succeeds on E.
(viii) E has a join tree (i.e., E is α-acyclic).
Proof. In Theorems 4.6 and 4.7, it is already shown that conditions (i), (vii) and
(viii) are equivalent.
(i) =⇒ (ii): Assume that E has the running intersection property. Let (e1 , e2 , . . . , em )
be an ordering of E such that for all i ∈ {2, . . . , m}, there is a ji < i such that
ei ∩ (e1 ∪ . . . ∪ ei−1 ) ⊆ eji . Now we show that (. . . ((e1 ⊲⊳ e2 ) ⊲⊳ e3 ) . . . ⊲⊳ em ) is
a monotone, sequential join expression: If r = {r1 , . . . , rm } is a pairwise consistent
database over E = {e1 , e2 , . . . , em }, then the join r1 ⊲⊳ . . . ⊲⊳ ri (which we abbreviate
as qi ) is consistent with ri+1 (1 ≤ i < n).
An easy inductive argument shows that rk = qi [ek ] whenever k ≤ i. In particular, let
k = ji+1 , and let V := ei+1 ∩ (e1 ∪ . . . ∪ ei ). Since V ⊆ em , it follows that rk [V ] = qi [V ].
But also ri+1 [V ] = rk [V ] since ri+1 and rk are consistent. Hence ri+1 [V ] = qi [V ]. So
ri+1 is consistent with qi which was to be shown.
(ii) =⇒ (iii): This is immediate since every monotone sequential join expression is a
monotone join expression.
(iii) =⇒ (iv): Assume that E has a monotone join expression. We must show that
every pairwise consistent database over E is globally consistent. Let r be a pairwise
consistent database over E. It is not hard to see that since no tuples are lost in joining
together the relations in r as dictated by the monotone join expression, it follows that
every member of r is a projection of the final result ⊲⊳ r. Hence r is globally consistent,
which was to be shown.
(iv) =⇒ (v): Details are described in [28].
(v) =⇒ (i): Details are described in [28] - GYO reduction succeeds.
(iv) =⇒ (vi): Assume that every pairwise consistent database over E is globally consistent. Let r1 , . . . , rm be a database over E. We have to show that r1 , . . . , rm has
a full reducer, i.e., after finitely many semijoins ri ⋉ rj , we obtain a globally consistent database. Note that when further semijoin operations do not change anything,
46
the resulting relations are pairwise consistent. By assumption, these are also globally
consistent which means that we have a full reducer.
(vi) =⇒ (iv): Assume that every database over E has a full reducer. Let r1 , . . . , rm
be a pairwise consistent database over E. By assumption, it has a full reducer but in
the case of pairwise consistent relations, the input and output of the full reducer is the
same, i.e., the result of the full reducer is the database r1 , . . . , rm itself, and the result
of a full reducer is guaranteed to be globally consistent. Thus, r1 , . . . , rm is globally
consistent.
2
47
5
Dually Chordal Graphs
5.1
Dually Chordal Graphs, Maximum Neighborhood Orderings and Hypertrees
Theorem 3.4 says that a graph is chordal if and only if it has a clique tree, i.e., a graph
G is chordal if and only if its hypergraph C(G) of maximal cliques is α-acyclic (or coarboreal). The dual variant of this means that C(G) is a hypertree; the corresponding
graph class called dually chordal graphs was studied in [65, 189, 194] and has remarkable
properties. In particular, the notion of maximum neighbor and maximum neighborhood
ordering (used in [29, 60, 189]) has many consequences for algorithmic applications and
is somehow dual to the notion of a simplicial vertex. For the next definition, we recall
the notation of neighborhood in the remaining subgraph:
Let G = (V, E) be a graph and (v1 , . . . , vn ) be a vertex ordering of G. For all i ∈
{1, . . . , n}, let Gi := G[{vi , . . . , vn }] and Ni [v] be the neighborhood of v in Gi : Ni [v] :=
N[v] ∩ {vi , . . . , vn }.
Definition 5.1 Let G = (V, E) be a graph.
(i) A vertex u ∈ N[v] is a maximum neighbor of v if for all w ∈ N[v], N[w] ⊆ N[u],
i.e., N 2 [v] = N[u]. (Note that possibly u = v in which case v sees all vertices of
G.)
(ii) A vertex ordering (v1 , v2 , . . . , vn ) of V is a maximum neighborhood ordering of
G if for all i ∈ {1, . . . , n}, vi has a maximum neighbor in Gi , i.e., there is a
vertex ui ∈ Ni [vi ] such that for all w ∈ Ni [vi ], Ni [w] ⊆ Ni [ui ] holds.
(iii) A graph is dually chordal if it has a maximum neighborhood ordering.
Note that dually chordal graphs are not a hereditary class; adding a universal vertex
makes every graph dually chordal. The following characterization of dually chordal
graphs shows that these graphs are indeed dual (in the hypergraph sense) with respect
to chordal graphs:
Theorem 5.1 ([65, 194]) For a graph G, the following conditions are equivalent:
(i) G has a maximum neighborhood ordering.
(ii) There is a spanning tree T of G such that every maximal clique of G induces a
subtree of T .
(iii) There is a spanning tree T of G such that every disk of G induces a subtree of T .
(iv) N (G) is a hypertree.
48
(v) N (G) is α-acyclic.
Proof. Let G = (V, E) be a graph.
(i) =⇒ (ii): By induction on |V |. Let x be the leftmost vertex in a maximum neighborhood ordering of G and let y be a maximum neighbor of x, i.e., N 2 [x] = N[y]. If
x = y, i.e., N 2 [x] = N[x], then x sees all other vertices of G; let T be a star with central
vertex x which fulfills (ii). Now assume that x 6= y; by induction hypothesis, there is
a spanning tree of the graph G − x fulfilling (ii) for G − x. Among all such spanning
trees, choose a tree T in which y is adjacent to a maximum number of vertices from
N(x).
Claim 5.1.1 In T , y sees all vertices of N(x) \ {y}.
Proof of Claim 5.1.1. Assume to the contrary that there is a vertex z ∈ N(x) \ {y}
which is nonadjacent to y in T . Consider the T -path y − . . . − v − z connecting y and
z. Let Tv (Tz , respectively) be the connected component of T obtained by deleting
the T -edge vz such that Tv contains v (Tz contains z, respectively). Adding to these
subtrees Tv , Tz a new edge yz, we obtain the tree T ′ . Since y and z are adjacent in
G − x, T ′ is a spanning tree of G − x. Now we show that T ′ fulfills condition (ii) as
well.
Let Q be a maximal clique of G − x. If z ∈
/ Q then Q is completely contained in one of
the subtrees Tv , Tz , i.e., Q induces one and the same subtree in both T and T ′ . Now
suppose that z ∈ Q. Since N[z] ⊆ N[y] = N 2 [x], we have y ∈ Q by maximality of Q.
Let u1 , u2 be any two vertices of Q. If both belong to the same subtree Tv or Tz then
u1 and u2 are connected by the same path in T and T ′ , and we are done. Now let u1
be in Tv and u2 be in Tz . In Tv , the vertices y and u1 are connected by a T -path P1
consisting of vertices from Q. In a similar way, the vertices z and u2 are connected by
a T -path P2 in Tz . Gluing together these paths P1 and P2 with the edge yz, we obtain
a T ′ -path connecting u1 and u2 in T ′ . Hence any maximal clique Q of G − x induces a
subtree in T ′ , i.e., T ′ satisfies condition (ii) as well. This, however, contradicts to the
choice of T ; thus, in T , y sees all vertices of N(x) \ {y} which shows Claim 5.1.1.
Now let T be a spanning tree fulfilling the claim for G − x. Let T ∗ be the tree obtained
from T by adding a leaf x adjacent to y. Obviously, T ∗ fulfills condition (ii).
(ii) =⇒ (iii): Let T be a spanning tree of G such that every clique of G induces a subtree
in T . We claim that every disk N r [z] induces a subtree in T as well. In order to prove
this, it is sufficient to show that the vertex z and every vertex v ∈ N r [z] are connected
by a T -path consisting of vertices from N r [z]. Let v = v1 − v2 − . . . − vk − vk+1 = z be
a shortest G-path between v and z. By Qi we denote a maximal clique of G containing
the edge vi vi+1 , i ∈ {1, . . . , k}. From the choice of TS
, it follows that vi and vi+1 are
connected by a T -path Pi ⊆ Qi . The vertices of P = ki=1 Pi induce a subtree T [P ] of
T . Thus, v and z are connected by a T -path p. Since for all vertices w ∈ Qi , for the
G-distances d(z, w) ≤ d(z, vi ) ≤ r holds, every clique Qi is contained in the disk N r [z].
Thus, the claim follows from the obvious inclusion
49
p⊆P ⊆
k
[
Qi ⊆ N r [z].
i=1
(iii) =⇒ (iv) is obvious.
(iv) ⇐⇒ (v) is obvious by the self-duality of the neighborhood hypergraph N (G) and
the duality between hypertree and α-acyclicity.
(iv) =⇒ (i): Let N (G) be a hypertree. Then by Theorem 4.4, N (G) has the Helly
property and L(N (G)) is chordal. Let σ = (e1 , . . . , em ) be a perfect elimination ordering of L(N (G)). Since the hyperedges ei of N (G) are the closed neighborhoods,
σ = (N[v1 ], . . . , N[vn ]). Suppose inductively that there is a maximum neighborhood
ordering for G−v1 . It suffices to show that v1 has a maximum neighbor u1 . Since N[v1 ]
is simplicial in L(N (G)), the closed neighborhoods intersecting N[v1 ] are pairwise intersecting. By the Helly property of N (G), there is a vertex u1 in the intersection
of all such closed neighborhoods including N[v1 ] itself, i.e., there is a vertex u1 with
N 2 [v1 ] = N[u1 ]. Thus, u1 is a maximum neighbor of v1 .
The equivalence of (i) and (iv) can be shown in an easy direct way as follows: We know
already (iv) =⇒ (i). By Theorem 4.4, we can also show the other direction
(i) =⇒ (iv): Let G have the maximum neighborhood ordering σ = (v1 , . . . , vn ). We
have to show that N (G) has the Helly property and L(N (G)) is chordal.
Let N[x1 ], . . . , N[xk ] be a collection of pairwise intersecting closed neighborhoods in
G. Without loss of generality, let x1 be the leftmost vertex of x1 , . . . , xk in σ. Then x1
has aTmaximum neighbor u1 , i.e., there is a vertex u1 for which N 2 [x1 ] = N[u1 ]. Then
u1 ∈ ki=1 N[xi ], and thus, N (G) has the Helly property.
Now we show that N[v1 ] is simplicial in L(N (G)): Let N[x] and N[y] be closed neighborhoods intersecting N[v1 ]. Let v1 have a maximum neighbor u1, i.e., N 2 [v1 ] = N[u1 ].
Since x, y ∈ N 2 [v1 ], it follows that u1 ∈ N[x] ∩ N[y] and thus, N[v1 ] is simplicial in
L(N (G)). Inductively, it follows that σ = (N[v1 ], . . . , N[vn ]) is a perfect elimination
ordering of L(N (G)).
2
Since L(N (G)) is isomorphic to G2 (recall Proposition 2.5), Theorem 5.1 implies:
Corollary 5.1 Graph G is dually chordal if and only if G2 is chordal and N (G) has
the Helly property.
Another characterization which follows from the basic properties is the following:
Corollary 5.2 Graph G is dually chordal if and only if G = L(H) for some α-acyclic
hypergraph H.
Summarizing, we have:
Theorem 5.2 ([60, 65, 194]) Let G be a graph. Then the following are equivalent:
50
(i) G is a dually chordal graph.
(ii) N (G) is a hypertree.
(iii) C(G) is a hypertree.
(iv) G is the 2-section graph of some hypertree.
Thus, Theorems ?? and 5.2 together with (2.5) and the duality properties in Lemma
?? imply:
Corollary 5.3 ([60, 65, 194]) Let G be a graph and H be a hypergraph.
(i) G is dually chordal if and only if G2 is chordal and N (G) has the Helly property.
(ii) If H is α-acyclic then its line graph L(H) is dually chordal.
(iii) If H is a hypertree then its 2-section graph 2sec(H) is dually chordal.
As a corollary of Theorem 5.1, dually chordal graphs can be recognized in linear time
since α-acyclicity of N (G) can be tested in linear time [421]. Parts of Theorem 5.1
were found also by Szwarcfiter and Bornstein [418] and later again by Gutierrez and
Oubiña [256]; in particular, it was shown in [418] that dually chordal graphs are the
clique graphs of intersection graphs of paths in a tree. This implies that dually chordal
graphs are the clique graphs of chordal graphs (in the sense of Definition 2.10). See
[59, 60] for algorithmic applications of maximum neighborhood orderings and [93] for
more structural details. In [414], a linear time algorithm for constructing a special
(canonical) maximum neighborhood ordering for a dually chordal graph is described.
New characterizations of dually chordal graphs in terms of separator properties are
given by De Caria and Gutierrez in [177, 179]. Another new characterization was
found by Leitert in [323].
In [367], Moscarini introduced the concept of doubly chordal graphs, i.e., the graphs
which are chordal and dually chordal. This class was introduced for efficiently solving the Steiner problem (motivated by database theory); this can be done, however,
also for the larger class of dually chordal graphs (see [60]) and also for the class of
homogeneously orderable graphs which contain the dually chordal graphs [67]:
doubly chordal ⊂ dually chordal ⊂ homogeneously orderable
5.2
Bipartite Graphs, Hypertrees and Maximum Neighborhood Orderings
For bipartite graphs B = (X, Y, E), the one-sided neighborhood hypergraphs are of
fundamental importance. Let
51
N X (B) = {N(y) | y ∈ Y } as well as
N Y (B) = {N(x) | x ∈ X}.
Note that (N X (B))∗ = N Y (B) and vice versa.
Motivated by relational database schemes, the following concepts were introduced:
Definition 5.2 (Ausiello, D’Atri, Moscarini [11]) Let B = (X, Y, E) be a bipartite graph.
(i) B is X-conformal if for all S ⊆ Y with the property that all vertices of S have
pairwise distance 2, there is an x ∈ X with S ⊆ N(x).
(ii) B is X-chordal if for every cycle C in B of length at least 8, there is a vertex
x ∈ X which is adjacent to at least two vertices of C whose distance in C is at
least 4.
Analogously, define Y -conformal and Y -chordal for bipartite graphs.
These notions are justified by the following simple facts:
Proposition 5.1 ([11]) Let B = (X, Y, E) be a bipartite graph.
(i) B is X-conformal ⇐⇒ N Y (B) is conformal ⇐⇒ N X (B) has the Helly property.
(ii) B is X-chordal ⇐⇒ 2SEC(N Y (B)) is chordal ⇐⇒ L(N X (B)) is chordal.
Corollary 5.4 The following conditions are equivalent:
(i) B is X-chordal and X-conformal.
(ii) N Y (B) is α-acyclic.
(iii) N X (B) is a hypertree.
Maximum neighborhood orderings can be defined for bipartite graphs as well. For
this we need the following notations: Let B = (X, Y, E) be a bipartite graph, and let
(y1 , . . . , yn ) be a vertex ordering of Y . Then let BiY = B[X ∪ {yi , yi+1 , . . . , yn }] and let
Ni (x) denote the neighborhood of x in the remaining subgraph BiY .
Definition 5.3 Let B = (X, Y, E) be a bipartite graph.
(i) For y ∈ Y , a vertex x ∈ N(y) is a maximum neighbor of y if for all x′ ∈ N(y),
N(x′ ) ⊆ N(x) holds.
52
(ii) A linear ordering (y1 , . . . , yn ) of Y is a maximum X-neighborhood ordering of B
if for all i ∈ {1, . . . , n}, there is a maximum neighbor xi ∈ Ni (yi ) of yi :
for all x ∈ N(yi ), Ni (x) ⊆ Ni (xi ) holds.
Analogously, maximum Y -neighborhood orderings are defined.
Theorem 5.3 ([65]) Let B = (X, Y, E) be a bipartite graph. The following conditions
are equivalent:
(i) B has a maximum X-neighborhood ordering.
(ii) B is X-conformal and X-chordal.
Moreover, (y1 , . . . , yn ) is a maximum X-neighborhood ordering of B if and only if
(y1 , . . . , yn ) is a p.e.o. of 2SEC(N Y (B)).
Proof. (i) =⇒ (ii): Let σ = (y1 , . . . , yn ) be a maximum X-neighborhood ordering of
Y.
(a) B is X-conformal: Assume that the vertices in S ⊆ Y have pairwise distance 2.
Let yj ∈ S be the leftmost vertex of S in σ and let x be a maximum neighbor of yj in
BjY . Since every y ′ ∈ S has a common neighbor x′ ∈ X with yj , also x is adjacent to
y ′ which implies S ⊆ N(x). Thus, B is X-conformal.
(b) B is X-chordal: Let C = (xi1 , yi1 , . . . , xik , yik ), k ≥ 4, be a cycle in B. If C has
a chord then it has an X-vertex which fulfills the condition. Now assume that C is
a chordless cycle. Let yi1 = yj be the leftmost Y -vertex of C in (y1, . . . , yn ). Since
yik ∈ Nj (xi1 ) \ Nj (xi2 ) and yi2 ∈ Nj (xi2 ) \ Nj (xi1 ), the sets Nj (xi1 ) and Nj (xi2 ) are
incomparable with respect to set inclusion. Thus, neither xi1 nor xi2 are maximum
neighbors of yi1 . Let x be a maximum neighbor of yi1 = yj . Then yi1 , yi2 , yik ∈ Nj (x).
Thus, B is X-chordal.
(ii) =⇒ (i): Let B be X-conformal and X-chordal. Then by Proposition 5.1, the line
graph G′ = L(N X (B)) is chordal and N X (B) has the Helly property. Let (y1, . . . , yn )
be a p.e.o. of G′ . Thus NG′ [y1 ] is a a clique, i.e., for all y, y ′ ∈ NG′ [y1 ], N(y) ∩ N(y ′ ) 6=
∅. By the Helly property of N X (B), the total intersection of all N(y) such that
N(y) ∩ N(y1 ) 6= ∅ is nonempty: there is a vertex x ∈ X in all these neighborhoods.
Now, x is a maximum neighbor of y1 , and the same argument can be repeated with
the smaller graph B − y1 .
2
Corollary 5.5 Let B = (X, Y, E) be a bipartite graph. The following conditions are
equivalent:
(i) B has a maximum X-neighborhood ordering.
(ii) N X (B) is a hypertree.
53
(iii) N Y (B) is α-acyclic.
Theorems 5.1 and 5.3 imply the following connection between maximum neighborhood
orderings in graphs and in bipartite graphs.
Corollary 5.6 ([65]) A graph G has a maximum neighborhood ordering if and only if
B(G) has a maximum X-neighborhood ordering (maximum Y -neighborhood ordering,
respectively).
Proof. Recall that by Proposition 2.6, B(G) is isomorphic to the bipartite incidence
graph of N (G). By Theorem 5.1, G has a maximum neighborhood ordering if and
only if N (G) is a hypertree. Now, it is easy to see that the underlying tree of N (G)
immediately leads to the fact that N X (B(G)) is a hypertree as well, and for symmetry
reasons the same happens for N Y (B(G)). Conversely, if N X (B(G)) is a hypertree then
the underlying tree immediately leads to an underlying tree for N (G).
2
5.3
Further Matrix Notions
As already mentioned, a hypergraph H = (V, E) can be described by its incidence
matrix. The notion of a hypertree (see Definition 4.4) is also close to what is called
subtree matrix in [350].
Definition 5.4
(i) A Γ matrix has the form
1 1
1 0
(ii) A subtree matrix is the incidence matrix of a collection of subtrees of a tree, i.e.,
it is a (0, 1)-matrix with rows indexed by vertices of a tree T and columns indexed
by some subtrees of T and with an entry of 1 if and only if the corresponding
vertex is in the corresponding subtree.
(iii) An ordered (0, 1)-matrix M is supported Γ if for every pair r1 < r2 of row indices
and c1 < c2 of column indices whose entries form a Γ, there is a row index r3 > r2
with M(r3 , c1 ) = M(r3 , c2 ) = 1. One says that row r3 supports the Γ.
Theorem 5.4 (Lubiw [350]) A (0, 1)-matrix is a subtree matrix if and only if it is
a matrix with supported Γ ordering.
54
Proof. “=⇒”: Let M be a subtree matrix for a collection S of subtrees of a tree
T . Pick a vertex r of T and order the vertices of T by decreasing distance from r
(breaking ties arbitrarily). The distance between r and a subtree S is the minimum
distance between r and any vertex from S. Also order the subtrees from S by decreasing
distance from r.
We claim that this is a supported Γ ordering of M, for suppose vertices v1 < v2 and
subtrees t1 < t2 form a Γ in M: For i ∈ {1, 2}, let ri be the vertex of ti closest to
r. Then r1 ≥ v2 since v2 is in t1 . We claim that r1 supports the Γ: Since r1 is in t1 ,
M(r1 , t1 ) = 1. We have to show that r1 is also in t2 , i.e., M(r1 , t2 ) = 1. If r1 = v1 or
r1 = r2 , we are done. Now suppose that r1 6= v1 and r1 6= r2 .
Since t1 < t2 , r2 is closer to r than r1 but t1 and t2 contain a common vertex v1 , and
thus also r1 is on the T path between v1 and r2 , i.e., r1 is in subtree t2 and supports
the Γ.
“⇐=”: If the ordered n × m matrix M is supported Γ, create a tree T on vertex set
{1, 2, . . . , n} by setting for i ∈ {1, 2, . . . , n − 1}
f (i) =
min{k | M(i, k) = 1} if there exists j > i, M(j, k) = 1
not defined
otherwise
and b(i) = max{j | M(j, f (i)) = 1} and creating the edges (i, b(i)) if f (i) exists and
(i, n) otherwise. We claim that T defined in this way is a tree: Since b(i) > i when f (i)
(and thus also b(i)) exists, and the edges (i, n) otherwise, T is obviously cycle-free, and
for the same reason, T is connected.
Finally, we show that each column of M is the incidence vector of a subtree of T .
It suffices to show that for i < j, M(i, k) = M(j, k) = 1 implies M(b(i), k) = 1: If
this were not true then f (i) < k and rows i, b(i) and columns f (i), k would form an
unsupported Γ in M.
2
Note that Theorem 5.4 gives a characterization of hypertrees and of α-acyclic hypergraphs in terms of a matrix property. This also gives corresponding characterizations
of chordal as well as of dually chordal graphs.
Definition 5.5 A (0, 1)-matrix M is doubly lexically ordered if the rows (columns,
respectively) form a lexicographically increasing sequence from top to bottom (from left
to right, respectively) where for rows (columns, respectively), the rightmost position
(lowest position, respectively) has highest priority.
Theorem 5.5 (Lubiw [350]) Every (0, 1)-matrix M can be doubly lexically ordered
by some suitable permutations of rows and columns.
Proof. Let M = (Mij ) be an m × n matrix. We form a m · n vector d(M) as follows:
The entries of M will be ordered with respect to i + j, and for the same i + j with
respect to j:
d(M) = (M11 , M21 , M12 , M31 , M22 , M13 , M41 , . . . , Mmn )
55
d1 d3 d6 · . . .
d2 d5 ·
d4 ·
·
...
dm·n
Claim 5.3.1 If two rows (columns, respectively) of M are permuted which do not
appear in lexical order then the result d(M) will be lexically larger (with highest priority
at dm·n ).
Proof of Claim 5.3.1.
Let k, l be row indices of M with k < l and the property that the k-th row is lexically
larger than the l-th row.
Let j ∈ {1, . . . , n} be the largest index for which Mkj 6= Mlj ; then Mkj > Mlj . After
permuting the k-th and l-th row, the part of d(M) which was Mlj becomes Mkj and
the parts on its right hand side do not change their value, and analogously for columns.
This shows Claim 5.3.1.
By Claim 5.3.1, an ordering of M which maximizes d(M), is a doubly lexical ordering
of M.
2
Theorem 5.5 holds also for other ordered matrix entries instead of {0, 1}.
There is an efficient way for finding a doubly lexical ordering: Let L := n+m+ number
of 1’s in a (0, 1)-matrix M.
Theorem 5.6 (Paige, Tarjan [387]) A doubly lexical ordering of an m × n matrix
M over entries {0, 1} can be determined in O(L log L) steps.
56
6
Totally Balanced Hypergraphs and Matrices
6.1
Totally Balanced Hypergraphs Versus β-Acyclic Hypergraphs
Fagin [203, 204] defined β-acyclic hypergraphs in connection with desirable properties of relational database schemes. Recall that for α-acyclic hypergraphs, edgesubhypergraphs are not necessarily α-acyclic.
Definition 6.1 (Fagin [203, 204]) A hypergraph H = (V, E) is β-acyclic if each of
its edge-subhypergraphs is α-acyclic, i.e., for all E ′ ⊆ E, E ′ is α-acyclic.
Fagin [203] gives a variety of equivalent notions of β-acyclicity in terms of certain
forbidden cycles in hypergraphs (one of them goes back to Graham [248]) which Fagin
in [203] shows to be equivalent.
Actually, β-acyclic hypergraphs appear under the name totally balanced hypergraphs
much earlier in hypergraph theory (as it will turn out in Theorems 6.2 and 6.4).
Definition 6.2 (Berge [32], Lovász [341]) Let H = (V, E) be a hypergraph.
(i) A sequence C = (v1 , e1 , v2 , e2 , . . . , vk , ek ) of distinct vertices v1 , v2 , . . . , vk and
distinct hyperedges e1 , e2 , . . . , ek is a special cycle (or chordless cycle or induced
cycle or, in [342], unbalanced circuit) if k ≥ 3 and for every i, 1 ≤ i ≤ k,
vi , vi+1 ∈ ei (index arithmetic is done modulo k) and ei ∩{v1 , . . . , vk } = {vi , vi+1 }.
The length of cycle C is k.
(ii) H is balanced if it has no special cycles of odd length k ≥ 3.
(iii) H is totally balanced if it has no special cycles of any length k ≥ 3.
Special cycles are called weak β-cycles by Fagin in [203], and a hypergraph is called
β-acyclic if it has no weak β-cycles. Actually, [203] mentions four other conditions and
shows that all five are equivalent.
We will see in Theorem 6.4 that a hypergraph is β-acyclic if and only if it is totally
balanced. Totally balanced hypergraphs are a natural generalization of trees.
Balanced hypergraphs are a natural generalization of bipartite graphs. See the monograph of Berge [33] for many properties and characterizations of balanced hypergraphs,
and in particular, the following ones:
Theorem 6.1 ([33, 34]) For a hypergraph H, the following conditions are equivalent:
(i) H is balanced
57
(ii) the vertex-subhypergraphs of H are 2-colorable.
(iii) the vertex- and edge-subhypergraphs of H have the Kőnig property.
By Theorem 2.4, we have: If a hypergraph H has no special cycles of length 3 then it
has the Helly property. Moreover, the dual of a special cycle of length 3 is a special
cycle of length 3. Thus, the dual H ∗ also has no special cycle of length 3.
Corollary 6.1 ([33]) Any hypergraph without special cycle of length 3 has the Helly
property. In particular, balanced hypergraphs have the Helly property and are conformal.
In this subsection, we focus on totally balanced hypergraphs.
Proposition 6.1 Let H = (V, E) be a totally balanced hypergraph. Then the following
holds:
(i) The dual H ∗ of H and any vertex- or edge-subhypergraph of H are totally balanced.
(ii) H has the Helly property.
(iii) L(H) is chordal.
Proof. Let H = (V, E) be a totally balanced hypergraph.
(i): By definition, it immediately follows that the dual H ∗ and any vertex- or edgesubhypergraph of a totally balanced hypergraph H is totally balanced.
(ii): See Corollary 6.1.
(iii) L(H) is a chordal graph since H contains no special cycle.
2
Recall that vertex-subhypergraphs of hypertrees are not necessarily hypertrees. The
next theorem gives a characterization of totally balanced hypergraphs in terms of hypertrees.
Theorem 6.2 (Ryser [406], Lehel [321]) A hypergraph H is totally balanced if and
only if every vertex-subhypergraph of H is a hypertree.
Proof. Let H = (V, E) be a hypergraph. By Proposition 6.1, we have:
(i) The dual of H and any vertex- or edge-subhypergraph are totally balanced.
(ii) H has the Helly property.
(iii) L(H) is a chordal graph.
58
Now by Theorem 4.4, H (and every vertex-subhypergraph of H) must be a hypertree
and vice versa.
2
Actually, Lehel [321] gives a complete structural characterization of totally balanced
hypergraphs in terms of certain tree sequences. Lehel’s result implies the following
characterization of totally balanced hypergraphs which was originally found by Brouwer
and Kolen [111] and nicely corresponds to the existence of simple vertices in strongly
chordal graphs.
Theorem 6.3 ([111, 321]) A hypergraph H is totally balanced if and only if every
vertex-subhypergraph H ′ has a vertex v (a so-called nested point) such that the hyperedges of H ′ containing v are linearly ordered by inclusion.
By simple duality arguments, the next theorem follows immediately from Theorem 6.2.
Theorem 6.4 (D’Atri, Moscarini [171]) A hypergraph H is totally balanced if and
only if H is β-acyclic.
6.2
Totally Balanced Matrices
Definition 6.3 Let Bk denote the k × k square (0, 1)-matrix consisting of rows with
exactly two consecutive 1’s beginning with 10 . . . 01, then 110 . . . and so on; 00 . . . 11 is
the last row.
For example, B4 is the following matrix:
1
1
0
0
0
1
1
0
0
0
1
1
1
0
0
1
Thus, a hypergraph is totally balanced if and only if its incidence matrix contains no
square submatrix Bk for k ≥ 3 (in any row and column order), and correspondingly
for balanced hypergraphs and odd k ≥ 3. Obviously, the dual of a balanced (totally
balanced, respectively) hypergraph is balanced (totally balanced, respectively).
Lubiw in [350] defines totally balanced matrices in the following way:
Definition 6.4
(i) For n ≥ 3, a cycle matrix is an n × n (0, 1)-matrix with no identical rows and
columns which has exactly two 1’s in each row and in each column such that no
proper submatrix has this property.
(ii) A totally balanced matrix is a (0, 1)-matrix with no cycle submatrices.
59
Recall Definition 5.4 for the notion of a Γ submatrix.
Definition 6.5 An ordered (0, 1)-matrix M is Γ-free if M has no Γ submatrix.
Theorem 6.5 ([7, 279, 349]) A (0, 1)-matrix is totally balanced if and only if it has
a Γ-free ordering.
This is shown in [350] as a consequence of the existence of doubly lexical orderings and
the following:
Observation 6.1 If a (0, 1)-matrix has a cycle submatrix then for any ordering of the
matrix there is a Γ submatrix (formed by a topmost, leftmost 1 of the cycle submatrix;
the other 1 in its row in the cycle; and the other 1 in its column in the cycle).
Theorem 6.6 ([350]) Any doubly lexical ordering of a totally balanced matrix is Γfree.
For example, the matrix B4 from Definition 6.3 has the following doubly lexical ordering
which is not Γ-free:
1
1
0
0
1
0
1
0
0
1
0
1
0
0
1
1
(resulting from the matrix B4 by first permuting rows 1 and 2 and then permuting
columns 3 and 4)
60
7
Strongly Chordal and Chordal Bipartite Graphs
7.1
Strongly Chordal Graphs
The subsequently defined strongly chordal graphs are an important subclass of chordal
graphs for many reasons. Originally, they were introduced by Farber [207] as a subclass
of chordal graphs for which the domination problem ([GT2] in [224]), which remains
NP-complete for chordal graphs and even for split graphs [44], can be solved efficiently.
Chang and Nemhauser [124, 132] independently studied the same class and also showed
that some problems such as domination can be solved efficiently. Later on, this has
been extended to larger classes and other problems (see e.g. [59, 60, 67, 127]).
The motivation from the database community is the fact that strongly chordal graphs
are the 2-section graphs of β-acyclic hypergraphs (as it will turn out in Theorem 7.3
as a consequence of Theorem 6.4).
7.1.1
Elimination Orderings of Strongly Chordal Graphs
Farber [207] defined strongly chordal graphs in terms of so-called strong elimination
orderings which are closely related to neighborhood matrices of these graphs:
Definition 7.1 Let σ = (v1 , . . . , vn ) be an ordering of the vertex set V of G. The
neighborhood matrix Nσ (G) (N(G) if σ is understood) of G is the n × n matrix with
entries
nij =
1 if vi ∈ N[vj ]
0 otherwise
Note that this matrix is symmetric and the main diagonal has values 1:
vi ∈ N[vj ] ⇐⇒ vj ∈ N[vi ]
(N(G) is the incidence matrix of the closed-neighborhood hypergraph N (G)).
The subsequent Definition 7.2 must be read as follows: If in the (0, 1) neighborhood
matrix of graph G, for i < j and k < ℓ, the entries in row i and column k, in row i and
column ℓ as well as in row j and column k are 1, then the entry in row j and column ℓ
must be 1 as well (i.e., rows i < j and columns k < ℓ do not form a Γ - see Definition
5.4).
Definition 7.2 ([207]) Let G = (V, E) be a graph.
(i) A vertex ordering (v1 , . . . , vn ) of G is a strong elimination ordering (st.e.o.) if for
all i, j, k and ℓ with i < j, k < ℓ for vk , vℓ ∈ N[vi ] holds: if vj ∈ N[vk ] then also
vj ∈ N[vℓ ].
61
(ii) G is strongly chordal if G has a st.e.o.
Obviously, every st.e.o. is also a p.e.o. (let i = k in condition (i)); thus, strongly
chordal graphs are chordal.
Observation 7.1 Let σ = (v1 , . . . , vn ) be an ordering of the vertex set V of graph G.
Then σ is a st.e.o. of G if and only if the corresponding neighborhood matrix Nσ (G)
is Γ-free.
Proof. Let (v1 , . . . , vn ) be a st.e.o. We consider the i-th and the j-th row as well as
the k-th and the l-th column of the matrix, i < j, k < l.
Figure 8 schematically indicates the selected rows and columns.
k
l
i
j
Figure 8: Selected rows and columns
If nik = 1, nil = 1 and njk = 1 then vk ∈ N[vi ], vl ∈ N[vi ], vj ∈ N[vk ] and thus also
vj ∈ N[vl ], i.e., njl = 1.
Conversely, if the submatrix
11
10
is forbidden then obviously (v1 , . . . , vn ) is a st.e.o.
2
Observation 7.1 describes an important matrix aspect of strongly chordal graphs. We
will also show that strongly chordal graphs are the hereditarily dually chordal graphs.
For this, we need the following notion:
Definition 7.3 (Farber [207]) Let G = (V, E) be a graph.
(i) A vertex v ∈ V is called simple if the set of closed neighborhoods {N[u] | u ∈
N[v]} is linearly ordered with respect to set inclusion.
(ii) A vertex ordering (v1 , . . . , vn ) of V is a simple elimination ordering (si.e.o.) if for
all i ∈ {1, . . . , n}, vi is simple in Gi = G[{vi , . . . , vn }].
It is easy to see that every simple vertex is also simplicial, i.e., whenever a graph has
a simple elimination ordering, it is chordal.
For proving Theorem 7.1, we need the following property:
62
Lemma 7.1 Let v be simple in G = (V, E) and u0 ∈ N[v] be a vertex with smallest
neighborhood N[u0 ]. Then also u0 is simple in G.
Proof. Assume that u0 is not simple. Then let x, y ∈ N[u0 ] be two vertices with
incomparable neighborhoods N[x], N[y]. Since {N[u] | u ∈ N[v]} is linearly ordered
with respect to ⊆, for all u ∈ N[v] N[u0 ] ⊆ N[u] holds, in particular for u = v,
N[u0 ] ⊆ N[v]. Thus, v has two neighbors with incomparable neighborhood which is a
contradiction.
2
Theorem 7.1 ([207]) A graph G has a st.e.o. if and only if every induced subgraph
of G contains a simple vertex.
Proof. “=⇒”: If G has a st.e.o. (v1 , . . . , vn ) then by Definition 7.2 every induced
subgraph of G has such an ordering. We show that v1 is simple.
Let vk , vl ∈ N[v1 ] with k < l and vj ∈ N[vk ] with 1 < j. By Definition 7.2, it follows
immediately that vj ∈ N[vl ]. Thus, N[vk ] ⊆ N[vl ], and v1 is simple (which means that
the st.e.o. is also a si.e.o.)
“⇐=”: Assume that every induced subgraph of G contains a simple vertex. We recursively construct a st.e.o. (v1 , . . . , vn ) of G as follows: For every 1 ≤ i ≤ n, choose in
Gi = G({vi , . . . , vn }) a simple vertex vi with smallest |Ni [vi ]|.
We claim that this ordering is a st.e.o. Since the vertex vi is simple in Gi , i.e., for their
neighbors from Ni [vi ], the neighborhoods are linearly ordered with respect to ⊆, we
have:
The vertices from Ni [vi ] appear in (v1 , . . . , vn ) in the same order (this follows by Lemma
7.1 for Gi ). Now, for i < j and k < l let vk , vl ∈ N[vi ] and vj ∈ N[vk ].
Case 1. i < k. Then vi is simple in Gi and vk , vl ∈ Ni [vi ] with k < l. Thus, Ni [vk ] ⊆
Ni [vl ] and therefore also vj ∈ N[vl ].
Case 2. i = k. In this case, the assertion is fulfilled since any simple vertex is simplicial.
Case 3. i > k. Then vk is simple in Gk and vi , vj ∈ Nk [vk ], i < j. Thus, Nk [vi ] ⊆
Nk [vj ]. From vl ∈ N[vi ], l > k, it follows that vl ∈ Nk [vi ], thus also vl ∈ Nk [vj ] and
finally vj ∈ N[vl ].
2
Corollary 7.1 The following conditions are equivalent:
(i) G is strongly chordal.
(ii) G has a si.e.o.
(iii) G is hereditarily dually chordal, i.e., every induced subgraph of G is dually
chordal.
(iv) N (G) is β-acyclic (i.e., by Theorem 6.4, totally balanced).
63
Proof. By Theorem 7.1, (i) and (ii) are equivalent.
For the equivalence of (ii) and (iii), assume first that G has a si.e.o. Then every
induced subgraph of G has a si.e.o. as well, and note that a si.e.o. is also a maximum
neighborhood ordering which means that every induced subgraph of G is dually chordal.
Conversely, let G be a hereditarily dually chordal graph. Let v1 have a maximum
neighbor u1 , i.e., the neighborhood of u1 is largest among all N[u], u ∈ N[v1 ]. Then
a straightforward discussion shows that also the subgraph of G induced by N[v1 ] − u1
has a maximum neighborhood ordering and so on which leads to a linear ordering of
neighborhoods with respect to v1 , i.e., v1 is simple. Now the same can be repeated for
G[{v2 , . . . , vn }] which shows the equivalence.
The equivalence of (iii) and (iv) is a simple consequence of Theorems 5.1 and 6.2. 2
The equivalence of (i) and (iv) has been obtained independently by Iijima and Shibata
[288]; they showed that a graph is sun-free chordal (see Theorem 7.4) if and only if its
neighborhood matrix is totally balanced.
7.1.2
Γ-Free Matrices and Strongly Chordal Graphs
Definition 7.2 implies a useful characterization of strongly chordal graphs by matrices.
Observation 7.1 leads to the fastest known recognition algorithms for strongly chordal
graphs by using doubly lexical orderings of matrices as given in Definition 5.5 (which
permute rows and columns in a suitable way)–see the subsequent Theorem 7.2 and
Corollary 7.2.
Example 7.1 Take the graph G from Figure 9 with vertices 1, . . . , 6, edges
12, 23, 25, 34, 35, 56 and vertex ordering σ1 = (1, 2, 3, 4, 5, 6).
6
5
1
2
3
4
net
Figure 9: The net
The adjacency matrix M of this graph corresponding to σ1 :
64
1
2
3
4
5
6
1
1
1
0
0
0
0
2
1
1
1
0
1
0
3
0
1
1
1
1
0
4
0
0
1
1
0
0
5
0
1
1
0
1
1
6
0
0
0
0
1
1
M is not Γ-free and not doubly lexically ordered (e.g., the third and fifth row together
with the second and fourth column form a Γ, and likewise the fourth and fifth row
together with the third and fourth column) and not doubly lexically ordered (e.g., the
third column from the left is larger than the fourth column).
A strong elimination ordering for G is σ2 = (1, 4, 6, 2, 3, 5). The adjacency matrix M ′
of G resulting from this ordering σ2 is as follows:
1
4
6
2
3
5
1
1
0
0
1
0
0
4
0
1
0
0
1
0
6
0
0
1
0
0
1
2
1
0
0
1
1
1
3
0
1
0
1
1
1
5
0
0
1
1
1
1
M ′ is doubly lexically ordered.
Theorem 5.5 holds also for other ordered matrix entries instead of {0, 1}.
Recall Theorem 5.6 for an efficient way for finding a doubly lexical ordering. An efficient
(but not linear-time) recognition of strongly chordal graphs results from the following
property:
Theorem 7.2 (Lubiw [350]) A graph G is strongly chordal if and only if any doubly
lexical ordering of its neighborhood matrix N(G) is Γ-free.
The connection to Γ-free matrices has been used by Paige and Tarjan in [387] as well
as by Spinrad in [413] (see also [414]) to design fast (but not linear time) recognition
algorithms for strongly chordal graphs.
Corollary 7.2 Recognition of strongly chordal graphs can be done in time O(m·log n).
It is an open problem whether strongly chordal graphs can be recognized in linear time.
Recall that a hypergraph is defined to be totally balanced if it contains no special cycle
(Definition 6.2), and recall Corollary 7.1; this has been expressed in terms of totally
balanced matrices.
65
Recall also (see Definition 6.4) that a (0, 1)-matrix M is totally balanced if M contains
no submatrix which is the vertex-edge incidence matrix of a cycle of length ≥ 3 of an
undirected graph.
Example 7.2 The vertex-edge incidence matrix of C3 with vertices v1 , v2 , v3 and edges
e1 = {v1 , v2 }, e2 = {v2 , v3 }, e3 = {v1 , v3 } is
e1
e2
e3
v1
1
0
1
v2
1
1
0
v3
0
1
1
By Corollary 7.1, G is strongly chordal if and only if its closed neighborhood hypergraph
N (G) is totally balanced. Thus, the next theorem is no surprise:
Theorem 7.3 ([207]) A graph G is strongly chordal if and only if its neighborhood
matrix N(G) is totally balanced.
7.1.3
Strongly Chordal Graphs as Sun-Free Chordal Graphs
Strongly chordal graphs have a variety of different characterizations, among them one
in terms of forbidden induced subgraphs. Recall that we say a vertex x sees a vertex
y if x is adjacent to y; otherwise we say x misses y.
Definition 7.4
(i) A k-sun is a chordal graph G with 2k vertices, k ≥ 3, whose vertex set is partitioned into two sets W = {w0 , . . . , wk−1} and U = {u0 , . . . , uk−1}, such that
U = {u0, . . . , uk−1} induces a cycle (the central set of the sun called central
clique of the sun if G[U] is a complete graph), W is a stable set and for all
i ∈ {0, . . . , k − 1}, wi sees exactly ui and ui+1 (index arithmetic modulo k ).
(ii) A complete k-sun is a k-sun where G[U] is a clique.
See for example Figure 10 for 3-sun and complete 4-sun.
As shown in [124, 207], the following holds:
Lemma 7.2 In a chordal graph, every k-sun contains a complete k ′ -sun for some
k ′ ≤ k.
Proof. We follow the proof given in [124]: Let U = {u1 , . . . , un } and W = {w1 , . . . , wn }
describe the partition of the vertex set of an n-sun G. Since G is chordal and the degree
of every vertex wi in G is 2, its two neighbors ui and ui+1 are adjacent. The proof is by
66
Figure 10: 3-sun and complete 4-sun
induction on n. If n = 3 and n = 4 then the claim is obviously fulfilled. Suppose that
n > 4 and that Lemma 7.2 holds for all suns on fewer than 2n vertices and suppose
that G is an n-sun which is not complete. Let u1 miss uj for some j with 1 < j < n.
Since u1 sees u2 and un , there exist k and l such that u1 sees uk and ul but misses up
for any p with k < p < l. In that case,
G′ := G[{u1 , uk , uk+1 , . . . , ul , wk , . . . , wl−1 ]
is a smaller sun for which U ′ := {uk , uk+1 , . . . , ul } and W ′ := {u1 , wk , wk+1, . . . , wl−1 }
gives the required partition. By induction, G′ (and hence G) contains a complete sun.
2
Lemma 7.3 (Brouwer, Duchet, Schrijver [112]) Let p ≥ 3 be an integer and let
G be a graph in which every cycle of length k, for 4 ≤ k ≤ 2p, has a chord. Then
N (G) has a special cycle Cp with p vertices and p hyperedges if and only if G has an
induced p-sun.
Proof. Assume that G = (V, E) is a graph such that every cycle in G of length k,
4 ≤ k ≤ 2p, has a chord. Clearly, if K is some induced subgraph of G then N (K)
is isomorphic to an induced partial subhypergraph of N (G). Thus, the ’if’ part of
Lemma 7.3 is easy to see; the induced p-sun in G leads to a special cycle in N (G) with
p vertices and p hyperedges.
The converse is proved by contradiction: Assume that G is sun-free and suppose to
the contrary that N (G) has a special cycle Cp with p vertices and p hyperedges. Thus,
by definition, there exists a set A = {a1 , . . . , ap } and a set B = {b1 , . . . , bp } with the
following properties:
(1) (a1 , N[b1 ], . . . , ap , N[bp ]) is a special cycle in N (G).
(2) N[bj ] ∩ A = {aj , aj+1 } for every j, 1 ≤ j ≤ p (index arithmetic modulo p).
(2) is clearly equivalent to
(3) For j ∈
/ {i, i + 1}, ai 6= bj and ai bj ∈
/ E.
(Note that so far, we do not know whether A ∩ B = ∅.)
67
Claim 7.1.1 If (v1 , v2 , . . . , vq ) is a cycle C of G (4 ≤ q ≤ 2p), then either v2 vq is a
chord of C or C has a chord of the form v1 vk for some k, 3 ≤ k ≤ q − 1.
The proof easily follows from the assumption that every cycle in G of length k, 4 ≤
k ≤ 2p, has a chord.
Claim 7.1.2 G contains an edge of the form ai aj (i 6= j).
Otherwise, by (1), A ∩ B = ∅. Thus (a1 , b1 , a2 , b2 , . . . , ap , bp ) is a cycle of length 2p in
G which by assumption must have a chord. Claim 7.1.1 together with (3) implies that
such a chord is an edge between two vertices from B, and since every cycle of length
k, for 4 ≤ k ≤ 2p, has a chord, it turns out that bk bk+1 is a chord of this cycle for each
k (1 ≤ k ≤ p). Hence A ∪ B induces a (chordal) subgraph of G which is a sun of order
p: B is the central set, and A is the stable set. The contradiction proves Claim 7.1.2.
Claim 7.1.3 If ai aj is an edge of G, then ai ai+1 is also an edge of G.
By symmetry, we may suppose i = 1. Let j be the smallest integer for which
a1 sees aj . If j > 2, the vertices a1 , b1 , a2 , aj are different. The sequence
(a2 , b2 , a3 , b3 , . . . , aj−1 , bj−1, aj ) is a walk not passing through a1 or b1 , by (3). This
walk induces a minimal path, say P , from a2 to aj (including a2 and aj ). By (3) and
the definition of j, the cycle (a1 , b1 , P ) with length ≥ 4 has no chord containing a1 .
Hence b1 must see aj (Claim 7.1.1), in contradiction with (3). So, j = 2.
Claim 7.1.4 (a1 , . . . , ap ) is a cycle of G.
Claim 7.1.4 is an easy consequence of the previous claim.
Claim 7.1.5 ai 6= bj for all i, j, i.e., A ∩ B = ∅.
Otherwise N[bj ] would contain ai−1 , ai and ai+1 (Claims 7.1.3 and 7.1.4), in contradiction with (2) (i.e., the definition of a special cycle).
Claim 7.1.6 G contains some edge bi bj .
Otherwise, A ∪ B would induce a p-sun with central set A and stable set B.
For obtaining the final contradiction, we observe that in Claim 7.1.6, i and j play a
symmetrical role. So, we may assume without loss of generality that G contains an
edge of the form b1 bj with j 6= 2 (the arguments for j = 2 are similar). Then G has
the following cycle:
(b1 , a2 , a3 , a4 , . . . , aj , bj )
68
and, by Claim 7.1.1, there is some edge b1 ai (3 ≤ i ≤ j) or the edge bj a2 must exist,
contradicting (3). This final contradiction finishes the proof of Lemma 7.3.
2
Now, Lemma 7.3 implies a simple proof for the subsequent characterization of strongly
chordal graphs as sun-free chordal graphs; the original proof by Farber was long and
complicated.
Theorem 7.4 (Farber [207]) A graph G is strongly chordal if and only if G is sunfree chordal.
Proof. Theorem 7.4 follows by Lemma 7.3 and Corollary 7.1.
2
A similar characterization of dually chordal graphs was obtained in [189]: Recall that
G is a Helly graph if its disk hypergraph D(G) is Helly. A subgraph S of G is isometric
if dS (x, y) = dG (x, y) for all vertices x and y of S.
Theorem 7.5 (Dragan [189]) A graph G is dually chordal if and only if G is a Helly
graph containing no isometric complete k-suns for k ≥ 4.
An odd chord vi vj in an even cycle (v1 , . . . , v2k ) is a chord with odd |i − j|.
Theorem 7.6 ([207]) A graph G is strongly chordal if and only if it is chordal and
every even cycle of length at least 6 in G has an odd chord.
Proof. Let G be a chordal graph. If every even cycle of length at least 6 has an odd
chord, then G contains no induced k-sun, k ≥ 3. Thus, by Theorem 7.4, G is strongly
chordal.
Conversely, we use Theorem 7.3: If G is strongly chordal then its neighborhood matrix
N(G) is totally balanced. If there is a cycle (v1 , v2 , . . . , v2k ) in G without odd chord
then the submatrix of N(G) consisting of the rows corresponding to v1 , v3 , . . . , v2k−1
and the columns corresponding to v2 , v4 , . . . , v2k is precisely the incidence matrix of a
cycle of length k. Consequently, G is not strongly chordal.
2
Finally, we give yet another characterization:
Corollary 7.3 (Farber [207]) Graph G is strongly chordal if and only if C(G) is
totally balanced.
Proof. By Theorems 5.1, 6.2 and Corollary 7.1, G is strongly chordal if and only if
C(G) is totally balanced.
2
It follows from the basic properties that G is strongly chordal if and only if G = L(H)
for some totally balanced hypergraph H.
Recall that for a clique tree T of G, the intersections Q ∩ Q′ of maximal cliques for
which QQ′ ∈ E(T ) form the minimal vertex separators in G. Let S(G) denote the
separator hypergraph of G. It can be considered as the first derivative of a chordal
graph. McKee [358] discusses this concept in detail.
69
Theorem 7.7 (Ma, Wu [351]) Graph G is strongly chordal if and only if G is
chordal and its separator hypergraph S(G) is totally balanced.
Yet another characterization of strongly chordal graphs was given in [169] and in [357].
Motivated by investigating leaf powers, a new characterization of strongly chordal
graphs in terms of clique arrangements and bad cycles is given in [374].
7.2
Chordal Bipartite Graphs
The most natural variant of chordality for bipartite graphs is the following:
Definition 7.5 (Golumbic, Goss [238]) A bipartite graph is chordal bipartite if
each of its cycles of length at least 6 has a chord.
In the terminology of Definition 3.7, this means that a bipartite graph is chordal
bipartite if and only if it is (6, 1)-chordal. Note that chordal bipartite does not mean
“chordal and bipartite” (as the name might suggest); if graph G is chordal and bipartite,
G is a forest, whereas C4 is chordal bipartite.
Thus, a better name for chordal bipartite graphs would have been weakly chordal
bipartite since the following holds (recall that graph G is weakly chordal if G and G
are hole-free, i.e., every cycle in G and in G of length at least 5 has a chord):
Proposition 7.1 Let B = (X, Y, E) be a bipartite graph. The following conditions are
equivalent:
(i) B is chordal bipartite.
(ii) B is triangle-free and hole-free.
(iii) B is weakly chordal and bipartite.
Proof. (i) ⇒ (ii): Holds by definition.
(ii) ⇒ (iii): B is hole-free by assumption. The complement graph B of the bipartite
graph is obviously hole-free since X and Y are cliques in B.
(iii) ⇒ (i): Obvious.
7.2.1
2
Tree Structure of Chordal Bipartite Graphs
Chordal bipartite graphs have various characterizations in terms of elimination orderings and tree structure properties of related hypergraphs; see e.g. Theorem 7.8 and
[93, 236] for more details. They are closely related to strongly chordal graphs.
70
Theorem 7.8 A bipartite graph B = (X, Y, E) is (6, 1)-chordal (i.e., chordal bipartite)
if and only if every induced subgraph B ′ of B is X-conformal, Y -conformal and Xchordal, Y -chordal.
Proof. “=⇒”: Let B = (X, Y, E) be bipartite (6, 1)-chordal. Then every induced
subgraph B ′ of B is also bipartite (6, 1)-chordal.
We first show that B ′ is X- and Y -chordal: If C is a cycle of length at least 8 in B ′
then C has a chord {x, y}, x ∈ X, y ∈ Y . Let x1 , x2 ∈ X be the neighbors of y in C
and let y1 , y2 ∈ Y be the neighbors of x in C. Let C1 , C2 denote the subcycles defined
by the chord {x, y} subdividing C. Without loss of generality, assume |C1 | ≤ |C2 |.
Moreover, assume without loss of generality that x2 , y2 are in C2 . Then y und y2 have
distance at least 4 in C and x is a neighbor of both vertices. Likewise, x and x2 have
distance at least 4 in C and y is a neighbor of both vertices. Thus, B ′ is X-chordal
and Y -chordal.
Now we show that B ′ is X-conformal and Y -conformal: Let S ⊆ Y be a vertex set with
pairwise distance 2 in B ′ . We show inductively the existence of a vertex x ∈ X with
S ⊆ N(x): For |S| = 2 and |S| = 3, the assertion is obviously fulfilled (for |S| = 3, the
existence of a chord in any cycle of length 6 is used).
Now, by induction hypothesis, let the assertion be fulfilled for all S ′ , |S ′ | ≤ k, with
pairwise distance 2 and let S ⊆ Y , |S| = k + 1 be a vertex setwith pairwise
distance
k+1
k+1
2. Then for every k-elementary subset Si ⊆ S, i ∈ {1, . . . , k } (note k = k + 1)
there is a vertex xi for which Si ⊆ N(xi ). If there is an i with S ⊆ N(xi ) then the
assertion is fulfilled. Otherwise, we can assume that the vertices x1 , . . . , xk+1 have
exactly one nonneighbor in S: Without loss of generality, let
xi ∈
/ N(yi+2( mod k+1) )
Now there is a C6 (x1 , y1 , x2 , y3, xk , y4 ) which is a contradiction. Thus, there is an index
i such that S ⊆ N(Xi ). Analogously, one shows Y -conformality of B ′ .
“⇐=”: If every induced subgraph of B is X-conformal, Y -conformal and X-chordal,
Y -chordal then B cannot contain chordless cycles of length at least 6 since chordless
cycles of length 6 are neither X-conformal nor Y -conformal and chordless cycles of
length at least 8 are neither X-chordal nor Y -chordal.
2
By Corollary 5.4, Theorem 7.8 implies
Corollary 7.4 A bipartite graph B = (X, Y, E) is (6, 1)-chordal (i.e., chordal
bipartite) if and only if N X (B) and N Y (B) are β-acyclic (i.e., totally balanced).
The hypergraph properties also imply:
Lemma 7.4 A bipartite graph B is chordal bipartite if and only if B is the bipartite
incidence graph of a totally balanced hypergraph.
71
Proof. “=⇒”: Let B = (X, Y, E) be a bipartite (6, 1)-chordal graph. By the definitions of N X (B) and of the bipartite incidence graph of a hypergraph, it is easy to
see that B = I(N X (B)) holds. Now by Theorem 7.8, every induced subgraph B ′ of
B is X-conformal, Y -conformal and X-chordal, Y -chordal, and thus by Corollary 5.4,
N X (B ′ ) is a hypertree which by Theorem 6.2 implies that N X (B) is totally balanced.
“⇐=”: Let H = (V, E) be a totally balanced hypergraph and let B = I(H). If B
would contain a chordless cycle C6 , say (x1 , e1 , x2 , e2 , x3 , e3 ) with xi ∈ V and ei ∈ E
then (e1 , e2 , e3 ) would form a special cycle of length 3 in H which is impossible. If B
would contain a longer chordless cycle C2k , k ≥ 4, say, (x1 , e1 , x2 , e2 , . . . , xk , ek ) then
this would contradict the chordality of L(H) (see Proposition 6.1).
2
Theorems 5.3 and 7.8 imply:
Corollary 7.5 (Dragan, Voloshin [195]) A bipartite graph B = (X, Y, E) is
chordal bipartite if and only if every induced subgraph of B has a maximum Xneighborhood ordering and a maximum Y -neighborhood ordering.
7.2.2
Relationship between Strongly Chordal and Chordal Bipartite
Graphs
Strongly chordal graphs are closely related to chordal bipartite graphs: Let BC (G)
denote the bipartite incidence graph I(C(G)).
Lemma 7.5 (Farber [207]) A graph G is strongly chordal if and only if BC (G) is
chordal bipartite.
Proof. Lemma 7.5 is an obvious consequence of Theorem 7.8 and Corollary 7.3.
2
For a bipartite graph B = (X, Y, E), let splitX (B) denote the one-sided completion
when X becomes a clique. Another relation between chordal bipartite and strongly
chordal graphs is the following (see also [65]):
Lemma 7.6 (Dahlhaus [166]) A bipartite graph B = (X, Y, E) is chordal bipartite
if and only if splitX (B) is strongly chordal.
Lemma 7.6 is a simple consequence of the following more general property:
Proposition 7.2 ([65]) Let B = (X, Y, E) be a bipartite graph. Then
(i) N X (B) has the Helly property if and only if C(splitX (B)) has the Helly property.
(ii) L(N X (B)) is chordal if and only if L(C(splitX (B))) is chordal.
72
7.2.3
Edge Elimination Orderings of Chordal Bipartite Graphs
Another aspect of bipartite graphs related to tree structure is the notion of edge elimination orderings.
Definition 7.6 Let B = (X, Y, E) be a bipartite graph. Then uv ∈ E is a bisimplicial
edge if N(x) ∪ N(y) induces a complete bipartite subgraph in B.
Obviously, an edge xy ∈ E is bisimplicial in B if and only if it is not the mid-edge of
any P4 in B.
Let (e1 , . . . , ek ) be an ordering of pairwise vertex-disjoint edges of B (not necessarily
all edges of E). Let Si be the set of endpoints of the edges e1 , . . . , ei and let S0 = ∅.
Definition 7.7 (Golumbic, Goss [238]) Let B = (X, Y, E) be a bipartite graph.
(i) (e1 , . . . , ek ) is a perfect edge elimination ordering (p.e.e.o.) for B if B((X ∪ Y ) \
Sk ) has no edge and each edge ei is bisimplicial in the remaining induced subgraph
B((X ∪ Y ) \ Si−1 ).
(ii) B is a perfect elimination bipartite graph if B admits a p.e.e.o.
Note that not every perfect elimination bipartite graph is chordal bipartite as the
example of the 6-pan shows, and the property of being perfect elimination bipartite is
not hereditary.
Recall the definition of bipartite copy B(G) = (V ′ , V ′′ , F ) of G - see Definition 2.9 - and
the fact that B(G) is isomorphic to the bipartite incidence graph I(N (G)). The next
theorem gives a connection between chordal graphs and perfect elimination bipartite
graphs.
Theorem 7.9 (Brandstädt [55]) A graph G is chordal if and only if B(G) is perfect
elimination bipartite.
Moreover:
Theorem 7.10 (Golumbic, Goss [238]) Let B = (X, Y, E) be a bipartite graph.
The following conditions are equivalent:
(i) B is chordal bipartite.
(ii) Every induced subgraph of B is perfect elimination bipartite.
(iii) Every minimal edge separator in B induces a complete bipartite subgraph.
73
Golumbic and Goss [238] gave an O(n5 ) recognition algorithm for perfect elimination
bipartite graphs which was improved by Goh and Rotem [235] to O(n3 ). Both of these
algorithms are based directly on the definition and the fact that a greedy algorithm
can be used to construct the perfect edge elimination ordering.
Another connection between strongly chordal graphs and chordal bipartite graphs is
given in the following lemma:
Lemma 7.7 ([55]) A graph G is strongly chordal if and only if B(G) is chordal bipartite.
Proof. Lemma 7.7 is an obvious consequence of Corollary 7.1, Corollary 5.6, Corollary
5.5, and Theorem 7.8.
2
There is a similar edge elimination ordering where only the edges but not their endvertices are deleted. We call these orderings perfect edge-without-vertex elimination
orderings (p.e.w.v.e.o.).
Definition 7.8 Let B = (X, Y, E) be a bipartite graph and σ = (e1 , . . . , em ) an edge
ordering. Let Bi = (X, Y, Ei ) with E0 = E and Ei = Ei−1 \ {ei }. The ordering σ is a
perfect edge-without-vertex elimination ordering (p.e.w.v.e.o.) if for all i ∈ {1, . . . , m}
ei is bisimplicial in Bi−1 .
The following result was found by several researchers (among them Tse-Heng Ma
and Haiko Müller):
Theorem 7.11 A bipartite graph is chordal bipartite if and only if it admits a
p.e.w.v.e.o.
Proof. “=⇒”: Let B = (X, Y, E) be a chordal bipartite graph. For finding a
p.e.w.v.e.o. of B, we use Lemma 7.6 for splitX (B) and the fact that, as a strongly
chordal graph, splitX (B) has a si.e.o., and in such an ordering, the closed neighborhoods are linearly ordered. Thus, if the si.e.o. of splitX (B) starts with y1 ∈ Y then
for the leftmost neighbor x of y1 , the edge y1 x ∈ E is bisimplicial in B (and its removal makes the neighborhood of y1 smaller), and in the same way, one can continue
constructing a p.e.w.v.e.o. of B.
“⇐=”: Let σ = (e1 , . . . , em ) be a p.e.w.v.e.o. of B. If B contains a chordless cycle C
isomorphic to C2k , k ≥ 3, then let ei be the leftmost edge of C in σ. Then obviously,
ei is not bisimplicial in Bi which is a contradiction.
2
The greedy algorithm, taking any bisimplicial edge to start the ordering scheme, can be
used to construct a p.e.w.v.e.o. ordering. Kloks and Kratsch [307] give an efficient
algorithm for finding a p.e.w.v.e.o. of a given chordal bipartite graph which uses the
Γ-free ordering of the associated bipartite adjacency matrix.
74
7.2.4
Recognition and Applications of Chordal Bipartite Graphs
Spinrad [414] gives simple direct proofs of Lemmas 7.7 and 7.6 in terms of Γ-free
matrices and discusses the relationship between fast recognition of strongly chordal
graphs and fast recognition of chordal bipartite graphs; linear time for recognizing
chordal bipartite graphs would imply linear time for recognizing strongly chordal graphs
but not vice versa (a linear time algorithm for recognizing chordal bipartite graphs as
claimed in [427] turned out to contain a flaw). Linear-time recognition of strongly
chordal graphs (chordal bipartite graphs, respectively) is still an open problem. See
[285] for other characterizations of chordal bipartite graphs in terms of intersection
graphs of compatible subtrees.
In [41], a relationship between dismantlable concept lattices and chordal bipartite
graphs is shown. Let L(R) denote the concept lattice of R.
Theorem 7.12 ([41]) A concept lattice L(R) is dismantlable if and only if the associated bipartite graph bip(R) is chordal bipartite.
For a bipartite graph B = (X, Y, E), the bipartite complement (or mirror) of B has the
same two color classes x and Y , and for x ∈ X and y ∈ Y , xy is an edge in mir(B) if
and only if xy ∈
/ E.
Theorem 7.13 ([41]) The following are equivalent:
(i) L(R) and its mirror lattice L(R) are planar.
(ii) bip(R) is chordal bipartite and its mirror bipartite graph mir(bip(R)) is also
chordal bipartite.
A bipartite graph is auto-chordal-bipartite (an ACB graph for short) if B and mir(B)
are chordal bipartite. See [38] for the study of ACB graphs.
Proposition 7.3 ([38]) Let B = (X, Y, E) be a bipartite graph.
(i) B is an ACB graph if and only if B is (3K2 , C6 , C8 )-free if and only if splitX (B)
and splitX (B) are strongly chordal.
(ii) B is a P7 -free ACB graph if and only if splitX (B) and splitY (B) and are interval
graphs.
ACB graphs have unbounded Dilworth number. In [38], for every k, ACB graphs with
(bipartite) Dilworth number k are characterized.
75
8
Distance-Hereditary Graphs, Subclasses and γAcyclicity
8.1
γ-Acyclicity and Berge-Acyclicity
The notion of γ-acyclic hypergraphs is described by Fagin [203, 204]:
Definition 8.1 (Fagin [203, 204]) Let H = (V, E) be a hypergraph.
(i) A sequence C = (v1 , e1 , v2 , e2 , . . . , vk , ek ), k ≥ 3, of distinct vertices v1 , v2 , . . . , vk
and distinct hyperedges e1 , e2 , . . . , ek is a γ-cycle in H if for every i, 1 ≤ i ≤ k,
vi , vi+1 ∈ ei holds (index arithmetic modulo k) and for all i, 1 ≤ i < k, vi 6∈ ej
for j ∈
/ {i − 1, i} holds while vk ∈ ek−1 ∩ ek (and possibly in other hyperedges ei
as well).
(ii) A hypergraph is γ-acyclic if it has no γ-cycle.
Note that the only difference to special cycles is the condition 1 ≤ i < k instead of
1 ≤ i ≤ k, i.e., any special cycle is a γ-cycle. Thus, if a hypergraph has no γ-cycle then
it has no special cycle as well. A typical example of a γ-cycle is the following which
represents the maximal cliques of a gem: Let G = ({v1 , . . . , v5 }, E) be a gem with edges
E = {v1 v2 , v2 v3 , v1 v3 , v1 v4 , v3 v4 , v2 v5 , v3 v5 }, i.e., (v4 , v1 , v2 , v5 ) is a P4 and v3 is universal
in G; the maximal cliques are the triangles Q1 = G[{v1 , v2 , v3 }], Q2 = G[{v2 , v3 , v5 }]
and Q3 = G[{v1 , v3 , v4 }]. Then (v1 , Q1 , v2 , Q2 , v3 , Q3 ) is a γ-cycle of length 3.
Fagin [203] gives some other variants of γ-acyclicity and shows that all these conditions
are equivalent. A crucial property among them is the following separation property:
Theorem 8.1 A hypergraph H = (V, E) is γ-cyclic if and only if there is a nondisjoint
pair E, F of hyperedges such that in the hypergraph that results by removing E ∩F from
every edge, what is left of E is connected to what is left of F .
This leads to the following tree structure of separators in γ-acyclic hypergraphs (it
has been rediscovered under various names (such as laminar structure) in subsequent
papers on ptolemaic graphs, e.g. in [429]).
Definition 8.2 ([18, 203, 327, 438]) For a hypergraph H = (V, E), we define:
(i) Bachman(H) is the hypergraph obtained by closing E under intersection, i.e., S
is in Bachman(H) if it is the intersection of some hyperedges from H (including
the hyperedges from E themselves).
(ii) The Bachman diagram of H is the following undirected graph with Bachman(H)
as its node set, and with an edge between two nodes S, T if S is a proper subset
of T , i.e., S ⊂ T and there is no other W in Bachman(H) with S ⊂ W ⊂ T .
76
(iii) A Bachman diagram is loop-free if it is a tree.
The tree property of the Bachman diagram is closely related to uniqueness properties
in data connections; see [203] for a detailed discussion of various properties which
are equivalent to γ-acyclicity and related work on desirable properties of relational
database schemes.
The main theorem on γ-acyclicity is the following:
Theorem 8.2 ([203]) Let H = (V, E) be a connected hypergraph. The following are
equivalent:
(1) H is γ-acyclic.
(2) Every connected join expression over H is monotone.
(3) Every connected, sequential join expression over H is monotone.
(4) The join dependency ⊲⊳ H implies that every connected subset of H has a lossless
join.
(5) There is a unique relationship among each set of attributes for each consistent
database over H.
(6) The Bachman diagram of H is loop-free.
(7) H has a unique minimal connection among each set of its nodes.
Finally, a hypergraph H is called Berge-acyclic if its bipartite incidence graph I(H) is
acyclic [32].
Note that for the corresponding hypergraph classes, the following inclusions hold:
Berge-acyclic ⊂ γ-acyclic ⊂ β-acyclic ⊂ α-acyclic
8.2
Cographs
Cographs are an important subclass of distance-hereditary graphs which occurs in many
places; they are fundamental for clique-width. See [93] for additional information.
8.2.1
Cograph characterization
1 adds edges
For disjoint vertex sets A, B ⊆ V , the join operation (denoted by )
0
between all pairs x, y with x ∈ A, y ∈ B, and the co-join operation (denoted by )
adds nonedges between all pairs x, y with x ∈ A and y ∈ B. These notions are closely
related to connectedness of a graph and its complement: G is disconnected if and only
if G is decomposable into the co-join of two subgraphs, and G is disconnected if and
1
only if G is decomposable into the join of two subgraphs. Subsequently we also use 0 in order to denote the relationship between disjoint vertex sets.
and 77
Definition 8.3 Graph G is a cograph (complement-reducible graph) if G can be constructed from single vertices by a finite number of join and co-join operations.
Theorem 8.3 G is a cograph if and only if G is P4 -free.
Proof. “=⇒”: By induction on the number of vertices in G. For single vertices, the
1 2 and G1 , G2 being P4 -free. If G would
assertion is obviously true. Now, let G = G1 G
contain a P4 P = abcd then P has vertices from G1 and G2 . Assume first that P has
exactly one vertex from G1 . If a ∈ V (G1 ), b, c, d ∈ V (G2 ) then ac ∈
/ E contradicts to
the join between G1 and G2 , if b ∈ V (G1 ), a, c, d ∈ V (G2 ) then bd ∈
/ E contradicts to
the join between G1 and G2 . Now assume that P has exactly two vertices from each of
G1 , G2 . If a, d ∈ V (G1 ), b, c ∈ V (G2 ) then ac ∈
/ E contradicts to the join between G1
and G2 , if b, d ∈ V (G1 ), a, c ∈ V (G2 ) then ad ∈
/ E contradicts to the join between G1
and G2 , and if a, b ∈ V (G1 ), c, d ∈ V (G2 ) then ac ∈
/ E contradicts to the join between
G1 and G2 . In every case, there is a non-edge of the P4 between G1 and G2 , and thus
1 2 is again P4 -free.
G = G1 G
0 2 is again P4 -free if G1 and G2 are
In the same way one can show that G = G1 G
P4 -free.
“⇐=”: Let G be a P4 -free graph. We will show that then G is decomposable with re1 0 into subgraphs G1 , G2 , i.e., either G or G is disconnected.
spect to the operations ,
Assume that not every P4 -free graph would have this property. Then let G = (V, E)
be a smallest P4 -free graph not having this property, i.e., G is P4 -free connected and
co-connected but for every v ∈ V , either G−v is disconnected or G − v is disconnected.
Note that in this case, G has at least four vertices.
Case 1. G − v is disconnected. Let H1 , . . . , Hk , k ≥ 2, be the connected components
of G − v, i.e., there are no edges between Hi and Hj for i 6= j, i, j ∈ {1, . . . , k} but
since G is connected, v has edges to each of H1 , . . . , Hk . Let xi be a neighbor of v in
Hi . Since G is also co-connected, v has at least one non-neighbor in V \ {v}. Without
loss of generality, let y ∈ H1 be a vertex with vy ∈
/ E. Since H1 is connected, there is
a path Px1 y between x1 and y in H1 . Let x′ y ′ be the first edge on this path for which
vx′ ∈ E but vy ′ ∈
/ E holds. Since vx1 ∈ E, vy ∈
/ E, the existence of such an edge is
′
′
guaranteed. But now the vertices x2 , v, x , y induce a P4 in G - contradiction.
Case 2. The case that G − v is disconnected can be handled in the same way as the
previous Case.
2
Theorem 8.3 implies that the property of being a cograph is a hereditary property, i.e.,
if G is a cograph then every induced subgraph G′ of G is a cograph as well. Moreover,
it implies that cographs are weakly chordal (and thus perfect).
8.2.2
Tree Structure and Optimization on Cographs
The recursive generation of cographs by the two operations join and co-join is described
in a tree structure - the cotree. This tree has the vertices of the graph as its leaves,
78
1 and 0 according to the operations. If
and the internal nodes are labeled with 1 2 (G = G1 G
0 2 , respectively) then the root vertex of the cotree of G
G = G1 G
1 (,
0 respectively), and its two children are the root nodes of G1 , G2 ,
carries the label respectively.
A cotree is not necessarily a binary tree; for example, a clique with k vertices is
1 node with the k vertices as its children.
represented by one In [155], it is described how to recognize in linear time O(n + m) whether a given input
graph G is a cograph; starting with a single vertex, the algorithm tries to incrementally
construct a cotree T of G, i.e., in every step, a new vertex is added and the new cotree
is constructed if the graph is still a cograph; otherwise, an induced P4 in G is given
as output. The algorithm is performed by a complicated marking procedure which
cannot be described here. However, it has a remarkable property: It does not only give
the correct “Yes/No” answer to the recognition problem; if the answer is “Yes” then
the algorithm gives a certificate namely a cotree, and it is easily checkable whether
the cotree indeed represents the graph, and if the answer is “No”, it gives a certificate
for this answer, i.e., in the case of cograph recognition a P4 in the input graph. Such
recognition algorithms are called certified algorithms and are known for various graph
classes [313].
A simpler recognition of cographs is described in [175] whose time bound, however is
O(n + m log n) (and thus not linear). In [107], a simple multisweep LexBFS algorithm
for recognizing cographs in linear time is given.
Various algorithmic graph problems being NP-complete in general, can be solved
efficiently in a bottom-up procedure along the cotree of a cograph. As examples,
we describe this for the problems MAXIMUM STABLE SET and MAXIMUM
CLIQUE.
Let G = (V, E) be a graph. Recall the definition of independent sets and cliques in
section 2.
If G = (V, E) is a graph with vertex weight function w then for U ⊆ V , w(U) :=
Σx∈U w(x).
Let αw G be the maximum weight of a stable set in G, and let ωw G be the maximum
weight of a clique in G. Now, obviously the values of αw (G) and ωw (G) can be computed
1 2 and G = G1 G
0 2:
recursively for G = G1 G
1 2 then
• If G = G1 G
ωw (G) = ωw (G1 ) + ωw (G2 ) and
αw (G) = max(αw (G1 ), αw (G2 )).
0 2 then
• If G = G1 G
ωw (G) = max(ωw (G1 ), ωw (G2 )) and
αw (G) = αw (G1 ) + αw (G2 ).
79
This implies linear time algorithms for these two problems on cographs.
As a further example, we show how to color cographs in an optimal way.
Clearly, cographs, as a subclass of weakly chordal graphs are perfect. A direct proof is
given here:
Proposition 8.1 Cographs are perfect.
Proof. This is a consequence of the fact that cographs are distance hereditary, and
distance-hereditary graphs are perfect. A direct proof of Proposition 8.1 can be done
inductively on the number of vertices as follows: For one-vertex graphs, the claim
1 2 and ω(Gi ) = χ(Gi ) holds
is obviously fulfilled. Now assume first that G = G1 G
for i ∈ {1, 2}. Since there is a join between G1 and G2 , χ(G) = χ(G1 ) + χ(G2 ) =
ω(G1 ) + ω(G2) = ω(G) which shows the claim.
0 2 and ω(Gi ) = χ(Gi ) holds for i ∈ {1, 2}. Since there is a
Now assume that G = G1 G
co-join between G1 and G2 , χ(G) = max(χ(G1 ), χ(G2 )) = max(ω(G1 ), ω(G2)) = ω(G)
which again shows the claim. Thus, cographs are perfect.
2
Another remarkable property of cographs is the fact that they (and their complement)
1 2 , the edges
are transitively orientable (and thus permutation graphs): For G = G1 G
of the join are oriented from G1 to G2 - this obviously gives a transitive orientation if
it is assumed that G1 and G2 are already transitively oriented - and for a co-join, there
is nothing to show.
See [150, 154, 155, 93] for further properties of this graph class.
8.3
Distance-Hereditary Graphs
Distance-hereditary graphs are a fundamental generalization of trees. They are closely
related to γ-acyclic hypergraphs (see Definition 8.1) and have bounded clique-width.
Originally, they were defined via a distance property:
Definition 8.4 (Howorka [281]) A graph G = (VG , EG ) is distance hereditary if for
each connected induced subgraph F = (VF , EF ) of G, the distance functions dG in G
and dF in F coincide: For all x, y ∈ VF , dG (x, y) = dF (x, y).
Definition 8.5 A u-v-geodesic is a u-v-path α such that l(α) = dG (u, v). Let Φ be a
cycle of G. A path α is an essential part of Φ if α ⊂ Φ and l(Φ)/2 < l(α).
Theorem 8.4 (Howorka [281]) The following conditions are equivalent:
(i) G is distance hereditary.
(ii) Every induced path of G is geodesic.
(iii) No essential part of a cycle of G is induced.
80
(iv) Each cycle of G of length ≥ 5 has at least two chords, and each 5-cycle of G has
a pair of crossing chords.
(v) Each cycle of G of length ≥ 5 has a pair of crossing chords.
Proof. Howorka [281] has shown that (i) ⇐⇒ (ii) ⇐⇒ (iii) =⇒ (iv) =⇒ (v) =⇒ (iii);
here, we give his proof.
(i) =⇒ (ii): Let α be an induced path of a distance-hereditary graph G and let u and
v be the endpoints of α. Then dG (u, v) = dα (u, v) = l(α). Hence α is a geodesic.
(ii) =⇒ (i): Suppose that F is a connected induced subgraph of G. Let u, v be arbitrary
vertices of F , and let α be a u-v-geodesic of F . Thus naturally, α is an induced path
of F and, consequently, also an induced path of G. Hence, by assumption, α is a
u-v-geodesic of G. Thus dF (u, v) = l(α) = dG (u, v). This proves that G is distance
hereditary.
(ii) =⇒ (iii): Since an essential part of a cycle cannot be a geodesic, (ii) clearly implies
(iii).
(iii) =⇒ (ii): Let G be a graph satisfying (iii). Let u 6= v be vertices of G and assume
that α = (u = a0 , a1 , . . . , am = v) is a u-v-path of G which is not a geodesic. Consider
any u-v-geodesic β = (u = b0 , b1 , . . . , bn = v), n < m. Let i be the largest index for
which bi = ai , 0 ≤ i < n. Let t be the least index > i for which bt ∈ α. Thus bt = aj
for some j > t. Consequently, the path δ = (ai , ai+1 , . . . , aj ) is an essential part of the
cycle (ai , ai+1 , . . . , aj = bt , bt−1 , . . . , bi = ai ). By assumption, δ is not induced. Hence
α is not induced. This completes the proof.
(iii) =⇒ (iv): Let Φ = (a0 , a1 , . . . , an = a0 ), n ≥ 5, be a cycle of a graph G satisfying
(iii). By considering any essential part of Φ of length ≤ n − 2, we see from (iii) that
Φ must have at least one chord, say ai aj . Since, in turn, (ai+1 , ai+2 , . . . , ai−1 ) is an
essential part of Φ, then Φ must have a chord distinct from ai aj . This proves that each
cycle of G of length ≥ 5 has at least two chords. An easy verification shows that if
(iii) holds then a 5-cycle of G must have a pair of crossing chords.
(iv) =⇒ (v): Assume that G satisfies (iv). We will prove by induction that each n-cycle
of G, n ≥ 5, has a pair of crossing chords. By assumption, the assertion is true for
n = 5. Let n > 5 and suppose that each cycle of length m, 5 ≤ m < n, has a pair of
crossing chords. Consider an n-cycle Φ = (a0 , a1 , . . . , an = a0 ) and let ai aj and ar as
be two distinct chords of Φ. If they do not cross one another, we may assume without
loss of generality that 0 ≤ i ≤ j ≤ r ≤ s ≤ n. Consider the cycles (ai , aj , aj+1 , . . . , ai )
and (ar , as , as+1 , . . . , ar ). Since n ≥ 6, at least one of them has length ≥ 5 and hence,
by induction hypothesis, it must have a pair of crossing chords. This same pair is, of
course, a pair of crossing chords of Φ. This completes the proof.
(v) =⇒ (iii): Let G be a graph satisfying (v). We will prove by induction on n that
no essential part of an n-cycle of G is induced. This is trivially true if n = 3 or n = 4.
Assume that n > 4 and that the assertion is true for all cycles of length < n. Let
Φ = (a0 , a1 , . . . , an = a0 ) be a cycle of G and let α be an essential part of Φ, say
α = (a0 , a1 , . . . , ak ), where n/2 < k < n. Let ai aj and ar as be a pair of crossing chords
81
house
domino
gem
Figure 11: House, domino and gem are not distance-hereditary
of Φ. Without loss of generality we may assume that 0 ≤ i < j < n, 0 ≤ r < s < n
and i < r. If j ≤ k then ai aj joins two vertices of α; hence α is not induced. If
i ≥ k then α is an essential part of the cycle (a0 , a1 , . . . , ak , . . . , ai , aj , . . . , an = a0 )
of length < n. Hence, by induction hypothesis, α is not induced. We may assume
therefore that 0 ≤ i < k < j < n. Applying the same argument to ar as , we obtain
0 ≤ r < k < s < n. Since the chords ai aj and ar as cross one another, it follows that
0 ≤ i < r < k < j < s < n. Denote α′ = (a0 , a1 , . . . , ar ) and α′′ = (ai , ai+1 , . . . , ak ).
We claim that either α′ is an essential part of the cycle (as , as+1 , . . . , ar , as ) or α′′ is
an essential part of the cycle (ai , ai+1 , . . . , aj , ai ). We have indeed: l(α′ ) + l(α′′ ) ≥
k + 1 ≥ n − k + 2 > n − s + j − k + 2 = (n − s + 1) + (j − k + 1) and so, either
l(α′ ) > n − s + 1, or l(α′′ ) > j − k + 1, which proves our claim. It follows now from an
induction hypothesis that either α′ or α′′ is not an induced path. Hence α cannot be
induced. This completes the inductive step and proves the theorem.
2
Already in 1970, Olaru and Sachs ([407] Theorem 5) showed that graphs fulfilling
condition (v) of Theorem 8.4 are perfect.
Recall that holes are chordless cycles with at least five vertices. Obviously, holes are
not distance hereditary. Recall that, in connection with relational database schemes,
(k, l)-chordal graphs were defined (see Definition 3.7). The following characterization
of distance-hereditary graphs - see (ii) - was given in [23].
Theorem 8.5 Let G be a graph.
(i) G is (5, 2)-chordal if and only if G is (house, hole, domino)-free.
(ii) G is distance-hereditary if and only if G is (house, hole, domino, gem)-free.
Proof. (i): Obviously, every (5, 2)-chordal graph is (house, hole, domino)-free. For the
other direction, let G be (house, hole, domino)-free, and let C = (x1 , . . . , xk ), k ≥ 5,
be a cycle in G. If k = 5 then C is no C5 and no house, i.e., C must have at least two
chords. If k = 6 then C is no C6 and no domino, and since G is C5 -free, C must have
at least two chords. If k ≥ 7 then C has a chord xi xj since G is hole-free. A cycle C ′
consisting of an essential part of C together with the chord xi xj has length at least 5
and thus has another chord (since G is hole-free) which shows the assertion.
(ii): Obviously, every distance-hereditary graph is (house, hole, domino, gem)-free. For
the other direction, let G be (house, hole, domino, gem)-free, and let C = (x1 , . . . , xk ),
k ≥ 5, be a cycle in G. By Theorem 8.4, (v), it is sufficient to show that C has two
82
crossing chords. If k = 5 then C is no C5 , no house and no gem, i.e., C must have
two crossing chords. If k = 6 then C is no C6 and no domino, and since G is C5 - and
gem-free, C must have two crossing chords. If k ≥ 7 then C has a chord xi xj since G is
hole-free. A cycle C ′ consisting of an essential part of C together with the chord xi xj
has length at least 5 and thus, by an induction hypothesis, has two crossing chords
which shows the assertion.
2
Theorems 8.3 and 8.5 imply:
Corollary 8.1 Cographs are distance hereditary.
Since C5 = C5 and every anti-hole Ck , k ≥ 6, contains house as induced subgraph, it
follows:
Corollary 8.2 Distance-hereditary graphs are weakly chordal and thus perfect.
The next characterization of distance-hereditary graphs in terms of three simple operations is important for most of the algorithmic applications.
Theorem 8.6 (Bandelt, Mulder [23]) A connected graph G is distance-hereditary
if and only if G can be generated from a single vertex by repeatedly adding a pendant
vertex, a false twin or a true twin.
Proof. “⇐=”: Assume first that graph G can be generated from a single vertex by
repeatedly adding a pendant vertex, a false twin or a true twin. Then it can easily be
seen that G must be (house, hole, domino, gem)-free.
“=⇒”: For the other direction, we give the short proof of Theorem 8.6 contained in
[262]. Actually, [23] is claiming more namely that every distance-hereditary graph with
at least two vertices contains either a pair of twins or two pendant vertices. In [22],
an even slightly stronger version is given (and an incorrectness of the proof in [23] is
corrected).
Let G be a distance-hereditary graph, thus having crossing chords in each cycle of
length at least 5. It suffices to show that G has a pendant vertex or a pair of twins
since every induced subgraph of G is again distance hereditary. This is trivially fulfilled
if G is a disjoint union of cliques. We may assume that some component H of G is
not a clique. Let Q be a minimal cutset of H and R1 , . . . , Rm be the components of
H − Q. Suppose that |Q| ≥ 2; we show that Q is a homogeneous set. If not, there are
two vertices p, q ∈ Q and a vertex r ∈ V (H) − Q with rp ∈ E and rq 6∈ E. Let r ∈ R1 .
Since Q is a minimal cutset of H, vertex q has a neighbor s ∈ R1 . Note that there is
an r-s-path P1 in R1 . We choose s so that P1 is as short as possible. Similarly p has
a neighbor t ∈ R2 and q has a neighbor u ∈ R2 . We choose t and u so that a shortest
t-u-path P2 in R2 has smallest length (possibly t = u). The vertices s, q, u, t, p, r and
the paths P1 and P2 form a cycle C of length at least 5. The only possible chords of C
join p to q or to some vertices of P1 . Thus, C has no crossing chords, a contradiction.
83
Now if x is any vertex in R1 which is adjacent to Q, it must be adjacent to all vertices of
Q and thus Q is P4 -free (otherwise, G has a gem). By Theorem 8.3, a nontrivial P4 -free
graph has a pair of twins. They will also be twins in G because Q is homogeneous.
Now suppose that every minimal cutset contains only one vertex. Let R be a terminal
block of H, i.e., a maximal 2-connected subgraph of H that contains just one cutvertex, say x, of H. If |R| = 2, the vertex in R − x is a pendant vertex of G. If |R| ≥ 3
and R − x ⊆ N(x), the set R − x must induce a P4 -free subgraph. So R contains a
pair of twins, and clearly they are also twins in G.
If R \ N(x) 6= ∅, N(x) ∩ R is a cutset of H and so it contains a minimal cutset of size
one but then R is not 2-connected, a contradiction which proves Theorem 8.6.
2
For a distance-hereditary graph G, a pruning sequence of G describes how G can be
generated (dismantled, respectively) by repeatedly adding (deleting, respectively) a
pendant vertex, a false twin or a true twin. Pruning sequences and pruning trees are
a fundamental tool for most of the efficient algorithms on distance-hereditary graphs.
There is a more general way, however, to efficiently solve problems on graph classes
captured in the notion of clique-width described in the section on clique-width.
Definition 8.6 Let G be a graph with vertices v1 , . . . , vn , and let S = (s2 , . . . , sn )
be a sequence of tuples of the form ((vi , vj ), type), where j < i and type ∈
{leaf, true, f alse}. S is a pruning sequence for G, if for all i, 2 ≤ i ≤ n, the subgraph
of G induced by {v1 , . . . , vi } is obtained from the subgraph of G induced by {v1 , . . . , vi−1 }
by adding vertex vi and making it adjacent only to vj if type = leaf , making it a true
twin of vj if type = true, and making it a false twin of vj if type = f alse.
By Theorem 8.6, a graph is distance hereditary if and only if it has a pruning sequence.
Definition 8.7 Let G be a graph with vertices v1 , . . . , vn , and let S = (s2 , . . . , sn ) be
a pruning sequence for G. The pruning tree corresponding to S is the labeled ordered
tree T constructed as follows:
(1) Set T1 as the tree consisting of a single root vertex v1 , and set i := 1.
(2) Set i := i + 1. If i > n then set T := Tn and stop.
(3) Let si = ((vi , vj ), leaf ) (resp. si = ((vi , vj ), true), or si = ((vi , vj ), f alse)), then
set Ti as the tree obtained from Ti−1 by adding the new vertex vi and making it
a rightmost son of the vertex vj , and labeling the edge connecting vi to vj by leaf
(resp. by true or false).
(4) Go back to step (2) above.
A linear time recognition algorithm for distance-hereditary graphs using pruning sequences was claimed already in [262]; however, their algorithm contained a flaw.
Damiand, Habib and Paul [175] used the following characterization given by Bandelt
and Mulder for linear time recognition of distance-hereditary graphs.
84
Theorem 8.7 (Bandelt, Mulder [23]) Let G be a connected graph and L1 , . . . , Lk
be the distance levels of a hanging from an arbitrary vertex v of G. Then G is a distancehereditary graph if and only if the following conditions hold for any i ∈ {1, . . . , k}:
(i) If x and y belong to the same connected component of G[Li ] then Li−1 ∩ N(x) =
Li−1 ∩ N(y).
(ii) G[Li ] is a cograph.
(iii) If u ∈ Li and vertices x and y from Li−1 ∩ N(u) are in different connected
components X and Y of G[Li−1 ] then X ∪ Y ⊆ N(u) and Li−2 ∩ N(x) = Li−2 ∩
N(y).
(iv) If x and y are in different connected components of G[Li ] then sets Li−1 ∩ N(x)
and Li−1 ∩ N(y) are either disjoint or comparable with respect to set inclusion.
(v) If u ∈ Li and vertices x and y from Li−1 ∩ N(u) are in the same connected
component C of G[Li−1 ] then the vertices of C which are nonadjacent to u are
either adjacent to both x and y or to none of them.
The next theorem gives yet another characterization of distance-hereditary graphs. It
will be used in the following subsection.
Theorem 8.8 [23, 172] For a graph G, the following conditions are equivalent:
(1) G is distance-hereditary,
(2) For each vertex v of G and every pair of vertices x, y ∈ Li (v), that are in the
same connected component of the graph G[V \ Li−1 (v)], we have
N(x) ∩ Li−1 (v) = N(y) ∩ Li−1 (v).
Here, L1 (v), . . . , Lk (v) are the distance levels of a hanging from vertex v of G.
For many other graph classes defined in terms of metric properties in graphs, related
convexity properties and connections to geometry, see the recent survey by Bandelt
and Chepoi [21].
For distance-hereditary graphs, there is a long list of papers showing that certain
problems are efficiently solvable on this class. Most of these papers were published
before the clique-width aspect was found. Theorem 10.6 covers many of these problems; on the other hand, it might be preferable to have direct dynamic programming
algorithms using the tree structure of distance-hereditary graphs since the constant
factors in algorithms using Theorem 10.6 are astronomically large (and similarly for
graphs of bounded treewidth). However, various problems such as Hamilton Cycle
and variants cannot be expressed in MSOL; see also the algorithm for Steiner tree on
distance-hereditary graphs in section 8.4.
The four basic problems Maximum Independent Set, Maximum Clique, Chromatic
Number and Clique Cover can be solved in time O(n) if a pruning sequence of the
input graph is given [262]. In the next section we focus on the Steiner tree problem.
85
8.4
Minimum Cardinality Steiner Tree Problem in DistanceHereditary Graphs
For a given graph G = (V, E) and a set S ⊆ V (of target vertices), a Steiner tree
T (S, G) is a tree with the vertex set S ∪ S ′ (i.e., T (S, G) spans all vertices of S) and
the edge set E ′ such that S ′ ⊆ V and E ′ ⊆ E. The minimum cardinality Steiner tree
problem asks for a Steiner tree with minimum |S ∪ S ′ |.
An O(|V ||E|) time algorithm for the minimum cardinality Steiner tree problem
on distance-hereditary graphs was presented in [172]. Later, in [63], a linear time
algorithm was obtained as a consequence of a linear time algorithm for the connected
r-domination problem on distance-hereditary graphs. Here, we present a direct linear
time algorithm for the minimum cardinality Steiner tree problem.
Algorithm ST-DHG (Find a minimum cardinality Steiner tree in a distancehereditary graph)
Input: A distance-hereditary graph G = (V, E) and a set S ⊆ V of target vertices.
Output: A minimum cardinality Steiner tree T (S, G).
begin
pick an arbitrary vertex s ∈ S and build in G the distance levels L1 (s), . . . , Lk (s)
of a hanging from vertex s;
for i = k, k − 1, . . . , 2 do
if S ∩ Li (s) 6= ∅ then
find the connected components A1 , A2 , . . . , Ap of G[Li (s)];
in each component Aj pick an arbitrary vertex xj ;
order these components in non-decreasing order with respect to
d′ (Aj ) = |N(xj ) ∩ Li−1 (s)|;
for all components Aj taken in non-decreasing order with respect to
d′ (Aj ) do
set B := N(xj ) ∩ Li−1 (s);
if (S ∩ Aj 6= ∅ and S ∩ B = ∅) then
add an arbitrary vertex y from B to set S;
T (S, G) := a spanning tree of a subgraph G[S] of G induced by vertices S;
end
Clearly, this is a linear time algorithm. The correctness proof is based on Theorem
8.7, Theorem 8.8 and the following claims.
Let G = (V, E) be a distance-hereditary graph, S ⊆ V be a set of target vertices, and
s ∈ S be an arbitrary vertex from S.
Claim 8.4.1 There exists a minimum cardinality Steiner tree T (S, G) such that
dT (S,G) (x, s) = dG (x, s) for any vertex x of T (S, G).
86
Proof. Let L1 (s), . . . , Lk (s) be the distance levels of a hanging of G from vertex s ∈ S.
It is enough to show that there exists a minimum cardinality Steiner tree T (S, G) such
that if T (S, G) is rooted at s then for any vertex x of T (S, G) the following property
holds:
(P ∗ ) if x belongs to Li (s) (i ∈ {1, . . . , k}) then its parent x∗ in T (S, G) belongs to
Li−1 (s).
Let T (S, G) be a minimum cardinality Steiner tree with maximum number of vertices
satisfying property (P ∗ ) and let x be a vertex of T (S, G) not satisfying (P ∗) and with
maximum dG (x, s). Assume x belongs to Li (s). Consider the (x, s)-path P (x, s) in
T (S, G) and let y ∈ P (x, s) be the vertex closest to x in P (x, s) with y ∈ Li (s) and
y ∗ ∈ Li−1 (s), where y ∗ is the parent of y in T (S, G). From choices of vertices x and
y, we conclude that the subpath of P (x, s) between vertices x and y lays entirely in
Li (s) ∪ Li+1 (s). By Theorem 8.8, vertices x and y ∗ must be adjacent in G. Hence, we
can modify tree T (S, G) by removing edge xx∗ and adding edge xy ∗ . The new tree
obtained spans all vertices of S and has the same vertex-set. Since T (S, G) was chosen
to have maximum number of vertices satisfying property (P ∗ ), such a vertex x ∈ Li (s)
with x∗ ∈
/ Li−1 (s) cannot exist, proving the claim.
2
Let A1 , A2 , . . . , Ap be the connected components of G[Li (s)]. By Theorem 8.8, N(x) ∩
Li−1 (s) = N(y) ∩ Li−1 (s) for every pair of vertices x, y ∈ Aj , j ∈ {1, . . . , p}. Hence,
N(Aj ) ∩ Li−1 (s) = N(xj ) ∩ Li−1 (s) for any vertex xj ∈ Aj . Denote d′ (Aj ) := |N(Aj ) ∩
Li−1 (s)|. Assume, without loss of generality, that d′ (A1 ) ≤ d′ (A2 ) ≤ · · · ≤ d′ (Ap ). Let
Bj := N(u) ∩ Li−1 (s) = N(Aj ) ∩ Li−1 (s), where u is an arbitrary vertex of Aj .
Claim 8.4.2 For any vertices x, y ∈ Bj , N(x) \ (Bj ∪ A1 ∪ · · · ∪ Aj−1 ) = N(y) \ (Bj ∪
A1 ∪ · · · ∪ Aj−1 ).
Proof. We have u ∈ Li (s), x, y ∈ Li−1 (s) ∩ N(u) and every vertex of Aj is adjacent
to both x and y. By Theorem 8.8, any vertex z ∈ Li−2 (s) either adjacent to both x
and y or to none of them. Since d′ (Aj ) ≤ d′ (Aj ′ ) for j ′ > j, by Theorem 8.7(iv), any
vertex from Aj+1 ∪ · · · ∪ Ap = Li (s) \ (A1 ∪ · · · ∪ Aj ) is adjacent to both or neither one
of x and y. Assume now that there is a vertex z ∈ Li−1 (s) \ Bj which is adjacent to x
but not to y. Since path (z, x, u, y) lays in Li (s) ∪ Li−1 (s), by Theorem 8.8, there must
exist a vertex w in Li−2 (s) adjacent to all y, x, z. But then, it is easy to see that the
vertices u, x, y, z, w induce either a house or a gem in G, which is impossible.
2
Let now i be the largest number such that Li (s) ∩ S 6= ∅ and, as before, A1 , A2 , . . . , Ap
be the connected components of G[Li (s)] with d′ (A1 ) ≤ d′ (A2 ) ≤ · · · ≤ d′ (Ap ). Let
also j be the smallest number such that Aj ∩ S 6= ∅. Set B := N(Aj ) ∩ Li−1 (s). We
know that any vertex of Aj ∩ S is adjacent to all vertices of B.
Claim 8.4.3 Let S ∩ B 6= ∅, x ∈ S ∩ Aj and y ∈ S ∩ B. T ′ is a minimum cardinality
Steiner tree of G for target set S \ {x} if and only if T , obtained from T ′ by adding
vertex x and edge xy, is a minimum cardinality Steiner tree of G for target set S.
87
Proof. By Claim 8.4.1, for G and target set S, there exists a minimum cardinality
Steiner tree T where vertex x is a leaf and its neighbor x∗ in T belongs to Li−1 (s), i.e.,
to B. If x∗ 6= y, we can get a new minimum cardinality Steiner tree for G and target
set S by replacing edge xx∗ in T with edge xy. We can do that since vertex y is in T
and vertices x and y are adjacent in G.
2
Claim 8.4.4 Let S ∩ B = ∅, x ∈ S ∩ Aj , and y is an arbitrary vertex from B. T ′
is a minimum cardinality Steiner tree of G for target set S ∪ {y} \ {x} if and only if
T , obtained from T ′ by adding vertex x and edge xy, is a minimum cardinality Steiner
tree of G for target set S.
Proof. By Claim 8.4.1, for G and target set S, there exists a minimum cardinality
Steiner tree T such that dT (v, s) = dG (v, s) for any vertex v of T . In particular, vertex
x is a leaf and its neighbor x∗ in T belongs to Li−1 (s), i.e., to B. Furthermore, any
neighbor of x∗ in T must belong to Aj ∪ Aj+1 ∪ · · · ∪ Ap or to Li−2 (s). If x∗ 6= y, we can
get a new minimum cardinality Steiner tree for G and target set S by replacing in T
vertex x∗ with y and any edge ux∗ of T with edge uy. We can do that since, by Claim
8.4.2, vertex y is adjacent in G to every vertex u to which vertex x∗ was adjacent in T
(recall, u ∈ Aj ∪ Aj+1 ∪ · · · ∪ Ap ∪ Li−2 (s)).
2
Thus, we have:
Theorem 8.9 [63] The minimum cardinality Steiner tree problem in DistanceHereditary Graphs can be solved in linear O(|V | + |E|) time.
8.5
8.5.1
Further Important Subclasses of Distance-Hereditary
Graphs
Ptolemaic Graphs and Bipartite Distance-Hereditary Graphs
In this subsection, we describe the chordal and distance-hereditary graphs.
The ptolemaic inequality (∗) in metric spaces is defined as follows:
Definition 8.8 (Kay, Chartrand [299]) A connected graph G is ptolemaic if, for
any four vertices u, v, w, x of G,
(∗) d(u, v)d(w, x) ≤ d(u, w)d(v, x) + d(u, x)d(v, w).
Theorem 8.10 (Howorka [283]) Let G be a graph. The following conditions are
equivalent:
(i) G is ptolemaic.
(ii) G is distance hereditary and chordal.
88
(iii) G is gem-free and chordal.
(iv) For all distinct non-disjoint cliques P and Q of G, P ∩ Q separates P \ Q and
Q \ P.
The equivalence of (ii) and (iii) follows from Theorem 8.5: If G is distance-hereditary
then obviously G is gem-free. Conversely, if G is gem-free chordal then G is (house,
hole, domino, gem)-free and by Theorem 8.5, it is distance-hereditary.
Ptolemaic graphs are characterized in various other ways; see e.g. [429] where the
laminar structure of maximal cliques of ptolemaic graphs is described. This is closely
related to Bachman Diagrams as described in [203].
Recall that G is chordal if and only if C(G) is α-acyclic and G is strongly chordal if
and only if C(G) is β-acyclic. A similar fact holds for ptolemaic graphs (see Definition
8.1 for γ-acyclicity):
Theorem 8.11 ([171]) Graph G is ptolemaic if and only if the hypergraph C(G) of
its maximal cliques is γ-acyclic.
Theorems 8.5 and 8.4 imply:
Corollary 8.3 A graph is bipartite distance-hereditary if and only if it is bipartite
(6, 2)-chordal.
Proof. Obviously, bipartite (6, 2)-chordal graphs are (house, hole, domino, gem)-free
and thus, by Theorem 8.5, are distance-hereditary. Conversely, let G be a bipartite
distance-hereditary graph. Then, by Theorem 8.4, every cycle of length at least 5 has
two (crossing) chords which shows the assertion.
2
8.5.2
Block Graphs
There is an even more restrictive subclass of chordal distance-hereditary graphs, namely
the block graphs which can be defined as the connected graphs whose blocks (i.e., 2connected components) are cliques. Let K4 − e denote the clique of four vertices minus
an edge (also called diamond).
Buneman’s Four-Point Condition (∗∗) for distances in connected graphs requires that
for every four vertices u, v, x and y the following inequality holds:
(∗∗) d(u, v) + d(x, y) ≤ max {d(u, x) + d(v, y), d(u, y) + d(v, x)}.
It characterizes the metric properties of trees as Buneman [114] has shown:
Theorem 8.12 (Buneman [114]) A connected graph is a tree if and only if it is
triangle-free and fulfills the four-point condition (∗∗).
89
Theorem 8.13 (Howorka [282]) Let G be a connected graph. The following conditions are equivalent:
(i) G is a block graph.
(ii) G is (K4 − e)-free chordal.
(iii) G fulfills Buneman’s four-point condition (∗∗).
Theorem 8.14 (Berge [33]) G is a block graph if and only if C(G) is Berge-acyclic.
The following theorem appears as Theorem 3.5 in [265]:
Theorem 8.15 (Harary [265]) G is a block graph if and only if G is the intersection
graph of the blocks of some graph.
Theorem 8.5 in [265] says:
Theorem 8.16 A graph is the line graph of a tree if and only if it is a claw-free block
graph.
There are various other characterizations of block graphs - see e.g. [93] for a survey.
90
9
Tree Structure Decomposition of Graphs
Various kinds of decomposition of graphs such as modular decomposition and clique
separator decomposition lead to decomposition trees and algorithmic applications. In
this section, we first describe modular decomposition (cographs are the completely
decomposable graphs with respect to modular decomposition) and then mention some
other kinds of decompositions, and in particular clique separator decomposition.
9.1
Modular Decomposition of Graphs
Subsequently, the modular decomposition of arbitrary graphs is described which generalizes cographs and cotrees and gives a strong algorithmic tool for many problems.
See [355] for the connection between transitive orientation, cographs and modular decomposition.
9.1.1
Basic Module Properties
Let G = (V, E) be a graph. A vertex set M ⊆ V is a module in G if its vertices are
0
indistinguishable from outside M. More formally: For all u ∈ V \ M, either {u}M
1
or {u}M.
Sets A and B overlap if A \ B 6= ∅, B \ A 6= ∅, and A ∩ B 6= ∅.
Theorem 9.1 (Basic Module Properties [362]) Let G be a graph and let M(G)
denote the set of modules in G. Then the following properties hold:
(i) ∅, V and {v} for all v ∈ V are modules (the trivial modules);
(ii) If M1 , M2 ∈ M(G) then M1 ∩ M2 ∈ M(G);
(iii) If M1 , M2 ∈ M(G) and M1 ∩ M2 6= ∅ then M1 ∪ M2 ∈ M(G);
(iv) If M1 and M2 are overlapping modules then M1 \ M2 ∈ M(G), M2 \ M1 ∈ M(G),
(M1 \ M2 ) ∪ (M2 \ M1 ) ∈ M(G);
(v) If M is a module in G and U ⊆ V then M ∩ U is a module in G[U].
Proof. (i): This property is obviously fulfilled.
(ii): Let M1 and M2 be modules in G. If their intersection is empty then due to (i),
the assertion is fulfilled. Now assume that M1 ∩ M2 6= ∅. If M1 ⊆ M2 or M2 ⊆ M1
then again the assertion holds true. Now assume that M1 and M2 are overlapping
modules. Vertices outside M1 ∩ M2 cannot distinguish two vertices from M1 ∩ M2 : if a
vertex x ∈
/ M1 ∪ M2 would distinguish vertices a, a′ ∈ M1 ∩ M2 , i.e., xa ∈ E, xa′ ∈
/E
then this would contradict to the module property of M1 (M2 , respectively); if a vertex
x ∈ M1 \ M2 would distinguish vertices a, a′ ∈ M1 ∩ M2 , i.e., xa ∈ E, xa′ ∈
/ E then
this would contradict the module property of M2 , and the same holds for x ∈ M2 .
91
(iii): Let M1 ∩ M2 6= ∅ with a ∈ M1 ∩ M2 . If M1 ⊆ M2 or vice versa then the assertion
is trivial. Now assume that M1 and M2 are overlapping modules. Due to condition
(ii), vertices in M1 ∩ M2 cannot be distinguished from outside. The same holds for
two vertices in M1 (M2 , respectively). Now assume that vertices a′ ∈ M1 \ M2 and
a′′ ∈ M2 \ M1 could be distinguished by x ∈
/ M1 ∪ M2 : xa′ ∈ E and xa′′ ∈
/ E. Since
′
′
xa ∈ E and a, a ∈ M1 , also xa ∈ E holds but since a, a′′ ∈ M2 , it follows that
xa′′ ∈ E - contradiction. Thus M1 ∪ M2 is a module.
(iv): We first show that M1 \ M2 is a module. Assume to the contrary that there are
vertices a, a′ ∈ M1 \ M2 and x ∈
/ M1 \ M2 such that ax ∈ E, a′ x ∈
/ E. Then x ∈ M1
since M1 is a module, i.e., x ∈ M1 ∩ M2 . Let b ∈ M2 . Since x, b ∈ M2 and M2 is
a module, also ab ∈ E and a′ b ∈
/ E holds but now b ∈
/ M1 is a vertex outside M1
′
distinguishing vertices a, a ∈ M1 - contradiction. Analogously, M2 is a module.
Now we show that ∆ := (M1 \ M2 ) ∪ (M2 \ M1 ) is a module: Let a, a′ ∈ ∆. Since
M1 \M2 (M2 \M1 ) is a module, we can assume that a ∈ M1 \M2 and a′ ∈ M2 \M1 . Due
to (iii), M1 ∪ M2 is a module. Thus, a and a′ cannot be distinguished from outside
M1 ∪ M2 . Assume that there is a vertex x ∈
/ ∆, x ∈ M1 ∩ M2 such that xa ∈ E,
′
′
xa ∈
/ E. Since x, a ∈ M1 , a ∈
/ M1 and M1 a module, aa′ ∈
/ E holds. Since x, a′ ∈ M2 ,
a∈
/ M2 and M2 a module, aa′ ∈ E holds - contradiction.
(v): If M ⊆ U then the assertion is obviously fulfilled. Assume now that M \ U 6= ∅. If
M ∩ U = ∅ then again the assertion is obviously fulfilled. Now assume that M ∩ U 6= ∅
and M ∩ U is no module in G[U], i.e., there are vertices a, a′ ∈ M ∩ U and a vertex v ∈
U \M distinguishing a and a′ from outside M but then M is no module - contradiction.
2
Theorem 9.2 In a connected and co-connected graph G, the nontrivial ⊆-maximal
modules are pairwise disjoint.
Proof. Let M1 and M2 be nontrivial modules in G being maximal with respect to
set inclusion and assume that M1 ∩ M2 6= ∅. This implies that they are overlapping
modules. Then according to Theorem 9.1 (iii), M1 ∪ M2 is a module. If M1 ∪ M2 6= V
then M1 and M2 are not maximal - thus M1 ∪M2 = V . Note that vertices from M1 \M2
are either completely adjacent to M2 or completely nonadjacent to M2 , and the same
holds for vertices from M2 \ M1 . Let M1+ := {x : x ∈ M1 \ M2 and x has a join to M2 },
M1− := {x : x ∈ M1 \M2 and x has a cojoin to M2 , and define the sets M2+ and M2− in a
completely analogous way. Obviously, M1 \ M2 = M1+ ∪ M1− and M2 \ M1 = M2+ ∪ M2− .
If one of the sets M1+ , M1− , M2+ and M2− is empty then G is not connected or not
co-connected. Thus, all of these sets are nonempty. Now let x ∈ M1+ , x′ ∈ M1− and
y ∈ M2+ . The fact that xy ∈ E and M1 is a module implies that x′ y ∈ E but now x′
is adjacent to a vertex from M2 - contradiction.
2
A graph G is prime if G and G are connected and all of its modules are trivial. (In
various papers, a graph is called prime if all its modules are trivial but then P2 would be
a prime graph; alternatively, we can require that the graph has at least four vertices.)
92
Note that the smallest prime graph is the P4 . The characteristic graph G∗ of G is the
graph obtained by contracting the maximal modules of G to one vertex.
Theorem 9.3 The characteristic graph G∗ of a connected and co-connected graph G
is prime.
Proof. By Theorem 9.2, the maximal nontrivial modules in G = (V, E) are pairwise
disjoint and thus define a partition of V into equivalence classes. Let v ∗ denote the
equivalence class of a vertex v. Let G∗ = (V ∗ , E ∗ ) and U ⊆ V ∗ and denote by Kx the
equivalence class in V belonging to x ∈ V ∗ . Then the expansion E(U) of S
U is the union
of the equivalence classes belonging to U, i.e., the vertex set E(U) = x∈U Kx . We
first claim that for a module M in G∗ , its expansion E(M) is a module in G. Assume
to the contrary that there are a, b ∈ E(M) and x ∈
/ E(M) such that ax ∈ E and
bx ∈
/ E. Then obviously, a and b are not in the same class in E(M) since the classes
are modules. This means that a∗ 6= b∗ , a∗ , b∗ ∈ M and x∗ ∈
/ M but now M is no
module - contradiction. This shows the claim.
Now assume that M is a nontrivial module in G∗ . If M consists only of vertices whose
classes are one-elementary then E(M) = M and M is a module in G; thus, after
shrinking the modules in G, M cannot have more than one element. If M contains at
least one vertex u whose class U is a nontrivial module in G then U ⊂ E(M) but U
is a maximal module in G and E(M) is a module in G - contradiction. Thus, G∗ is a
prime graph.
2
9.1.2
Modular Decomposition Theorem
Theorems 9.1 and 9.2 lead to the following tree structure of a given graph G: Every
vertex in G is contained in a unique (possibly one-elementary) maximal module different from V , and these modules define a partition of V . The modular decomposition
tree has V as its root and the maximal modules smaller than V are the children of V
in the tree. Then the children of an inner vertex M are the maximal modules in G[M]
smaller than M. Thus, if the inner vertex M of the modular decomposition tree has
the partition M1 , M2 , . . ., Mk into its maximal modules then M1 , M2 , . . ., Mk are the
children of M. Note that the leaf descendants of M are the vertices of M, and the
edges in M between Mi and Mj are given by a sequence of join and co-join operations
between the modules Mi and the vertices outside Mi . The graphs being completely
decomposable by join and co-join are the cographs.
The following decomposition theorem is implicitly contained in the seminal paper [223]
by Tibor Gallai.
Theorem 9.4 (Modular Decomposition Theorem) Let G = (V, E) be an arbitrary graph. Then precisely one of the following conditions is satisfied:
(1) G is disconnected (i.e., decomposable by the co-join operation);
93
(2) G is disconnected (i.e., decomposable by the join operation);
(3) G and G are connected: There is some U ⊆ V and a unique partition P of V
such that
(a) |U| ≥ 4,
(b) G[U] is a maximal prime subgraph of G, and
(c) for every class S of the partition P, S is a module and |S ∩ U| = 1.
Each vertex of G forms a leaf of the decomposition tree. Each module M of G occuring
as a node in the tree contains exactly the vertices that are leaves of the subtree rooted
at M. According to the Decomposition Theorem, the tree has three kinds of nodes:
• parallel nodes (co-join operation);
• series nodes (join operation);
• prime nodes.
Linear time algorithms for finding the modular decomposition tree are given in [161,
170] and in [355]. See [422] for a simpler linear-time algorithm (as mentioned in [422],
the linear-time algorithm claimed in [258] contains an error that kills its simplicity).
The modular decomposition is of crucial importance in many algorithmic applications;
see [364] for many aspects of modular decomposition. Since for many algorithmic
problems the operations join and co-join are easy to handle (cf. the case of cographs),
it is important to look at prime graphs.
9.2
Prime extensions of subgraphs in prime graphs
If an induced subgraph in a prime graph G is not prime then it leads to larger subgraphs
in G which are prime. A crucial (and very useful) example is:
Figure 12: house, bull, and double-gem
Lemma 9.1 (Olariu [382]) If a prime graph contains an induced subgraph isomorphic to K3 then it contains house, bull or double-gem.
The following lemma (which is Claim 3.5 in [278]) gives the prime C4 extensions:
94
Figure 13: house, A, and domino
Lemma 9.2 (Hoàng, Reed [278]) If a prime graph contains an induced subgraph
isomorphic to C4 then it contains house, A or domino.
Proof. Let v0 , v1 , v2 , v3 with vi vi+1 ∈ E (addition modulo 4) induce a C4 in the prime
graph G = (V, E). Consider the subgraph H of G induced by the vertices that see
both v1 and v3 . Let H1 be the component in H containing v0 and v2 . Since G has no
homogeneous set, there exists a vertex x ∈
/ V (H1 ) distinguishing two vertices in H1 .
Since H1 is co-connected, x distinguishes two vertices w1 , w2 ∈ V (H1 ) with w1 w2 ∈
/ E,
say xw1 ∈ E and xw2 ∈
/ E. We know x ∈
/ V (H) because for all z ∈ V (H) \ V (H1 )
and all w ∈ V (H1 ), zw ∈ E, i.e., V (H1 ) is a module in H. Thus x sees at most one
of v1 and v3 . Now w1 , v1 , w2 , v3 induce a C4 in G. If x saw precisely one of v1 and v3
then x, w1 , v1 , w2 , v3 induce a house in G. Thus assume that x misses v1 and v3 . Now
consider the subgraph B induced by those vertices that see both w1 and w2 and miss x.
Let B1 be the component in B containing v1 and v3 . Since V (B1 ) is no homogeneous
set in G, there exists a vertex y that distinguishes two vertices u1 , u2 ∈ V (B1 ) with
u1 u2 ∈
/ E, say yu1 ∈ E and yu2 ∈
/ E. We know y ∈
/ V (B) since V (B1 ) is a module
in B. If y saw both w1 and w2 then, since y ∈
/ V (B), yx ∈ E but then x, y, w1, w2 , u2
induce a house in G. Thus assume that y sees at most one of w1 , w2. If y saw precisely
one of w1 , w2 then y, u1, u2 , w1 , w2 induce a house in G. In the other case when y sees
neither w1 nor w2 then x, y, w1 , w2, u1 , u2 induce an A or domino depending on whether
or not x sees y.
2
A similar lemma holds for extensions of diamond:
Figure 14: gem, d-A, and d-domino
Lemma 9.3 ([56, 78]) If a prime graph contains an induced subgraph isomorphic to
diamond then it contains d − A, d-domino or gem.
Of course, gem is not prime, and thus one needs another lemma giving the prime
extensions of gem. We do this here for its complement, the co-gem (see Figure 15):
95
Figure 15: Graphs P6 , net, A, xbull, X1 − X5
Lemma 9.4 ([87]) If a prime graph contains an induced subgraph isomorphic to cogem then it contains one of the following graphs: P6 , net, A, xbull, X1 , . . . , X5 .
Proof. Let g denote the (vertices of the) P4 of an induced co-gem in the prime graph
G = (V, E). Partition the vertex set V (G) − g into three sets:
1
A = {x ∈ V − g | xg},
0
B = {x ∈ V − g | xg},
C = V \ (g ∪ A ∪ B).
Note that B 6= ∅ because it contains the isolated vertex of the co-gem, and C 6= ∅
because otherwise g would be a homogeneous set in G. Suppose there is at least one
edge between B and C, say between b ∈ B and c ∈ C. All possible combinations of
edges between c and g lead to one of the graphs P6 , X1 , xbull, X2 , X3 , X4 , A, net from
0
Fig. 15. So we may assume that B C
for otherwise we are done. Since B 6= ∅ and
G is connected, there is a vertex b ∈ B that is adjacent to a vertex a ∈ A. Let c ∈ C
such that ca ∈
/ E. All possible combinations of edges between c and g contain one of
the graphs X3 , X4 , X5 from Fig. 15 as an induced subgraph. So partition set A into
two subsets
0
A0 = {x ∈ A | xB}
and
A1 = {x ∈ A | ∃b ∈ B : xb ∈ E}
1
and we may assume AC
for otherwise we are done. Note that for every c ∈ C there
exist non-adjacent vertices xc , yc ∈ g such that c is adjacent to xc and non-adjacent
to yc . In particular, there exists a connected component X of G[A ∪ C ∪ g] containing
C ∪ g. Then X ∩ A1 6= ∅ for otherwise X would be a homogeneous set in G. Now, an
X4 in G can be detected as follows. Consider a shortest path in X (recall that X is a
96
connected component in G) connecting a vertex in X ∩ A1 and a vertex in C (recall
that C 6= ∅). Let a1 a2 . . . ak c be such a path with a1 ∈ A1 and c ∈ C. Clearly, k ≥ 2
and ai ∈ A0 for all i = 2, . . . , k. Let b ∈ B be a neighbor (in G) of a1 . If k ≥ 5 then
a1 , . . . , a5 , b induce an X4 in G. If k = 4 then a1 , . . . , a4 , b, c induce an X4 in G. If
k = 3 then a1 , a2 , a3 , c, yc, b induce an X4 in G. Finally, if k = 2 then a1 , a2 , c, xc , yc , b
induce an X4 in G. This shows Lemma 9.4.
2
Lemma 9.4 was first shown in [79] using a rather complicated technique described in
[443] and in [78]; the direct proof given above is shorter.
Another interesting case is the set of claw extensions described in [79, 78].
Figure 16: Minimal prime extensions of claw
Lemma 9.5 ([78, 79, 444]) If a prime graph contains an induced subgraph isomorphic to claw then it contains at least one of the graphs in Figure 16 as an induced
subgraph.
A natural question asks about the finiteness of the set of prime extensions of a graph.
A first example is given in [78]: K4 has infinitely many minimal prime extensions;
see also [443]. In [233], a necessary and sufficient criterion is given when a graph has
infinitely many minimal prime extensions.
9.3
Graph classes with simple prime graphs
There are many graph classes where prime graphs with respect to modular decomposition have “simple structure”; the extreme case is the class of cographs where the
modular decomposition tree has no prime nodes.
9.3.1
P4 -Sparse Graphs
P4 -sparse graphs are a nice and fundamental example for a graph class having simple
prime graphs:
97
A graph G = (V, E) is P4 -sparse [275] if every five vertices induce at most one P4 in G.
Thus, cographs are P4 -sparse, and the only one-vertex extensions of a P4 in a P4 -sparse
graph G are the bull, gem and co-gem, i.e., G is P4 -sparse if and only if all the other
seven one-vertex extensions (such as P5 , C5 etc.) are forbidden induced subgraphs in
G:
Observation 9.1 A graph is P4 -sparse if and only if is (P5 , house, chair, co −
chair, P, co − P, C5)-free.
Obviously, the complement of a P4 -sparse graph is P4 -sparse.
A graph is a thin spider if its vertex set can be partitioned into a clique Q and a stable
set S such that the edges between Q and S form a matching, every vertex in S has
exactly one neighbor in Q, and at most one vertex in Q has no neighbor in S (the head
of the spider). Obviously, thin spiders are prime graphs and P4 -sparse. A graph is a
thick spider if it is the complement of a thin spider; it is a spider if it is a thin or thick
spider (these graphs were called turtles in Chinh T. Hoàng’s Master thesis [275]).
Theorem 9.5 (Hoàng [275]) A graph is P4 -sparse if and only if its prime induced
subgraphs are spiders.
Proof. Let G be a prime P4 -sparse graph. Assume that G contains C4 . Then by
Lemma 9.2, it would contain house, A or domino but each of them either is or contains
a forbidden induced subgraph (see Observation 9.1). The same argument holds if G
contains 2K2 , and anyway, G is C5 -free by assumption. Thus, by Theorem 3.11, we
have:
Claim 9.3.1 G is a split graph.
Let Q denote the clique and S denote the stable set in the split partition of G. We
claim that G is a spider: By Lemma 9.4, if prime P4 -sparse (split) graph G contains
co-gem then it contains net, say N with vertices x1 , x2 , x3 , y1 , y2 , y3 such that x1 , x2 , x3
is a clique, y1 , y2 , y3 is a stable set and the edges between x1 , x2 , x3 is a clique, y1 , y2 , y3
are xi yi , i = 1, 2, 3. We claim: that in this case, G is a thin spider:
Claim 9.3.2 If G contains co-gem then it is a thin spider.
Proof. If x ∈ Q then it is not a 1- or 2-vertex and if it is a 4- or 5-vertex for N then
there is a co-chair. Likewise, if x ∈ S then it is not a 3-, 4- or 5-vertex and if it is a 1or 2-vertex for N then there is a chair. Thus, N can have only 0-, 3- and 6-vertices; let
Z denote the set of 0-vertices, let M denote the set of 3-vertices (note that M ⊂ Q)
and let U denote the set of 6-vertices of N. Let Z1 := {z ∈ Z | z has a neighbor in M}
1
1 1 (else there is a chair in G)
and let Z0 := Z \ Z1 . Then obviously U M
and U Z
which implies that U ∪ Z0 = ∅ (else V \ (U ∪ Z0 ) is a homogeneous set in G. Now it
98
is clear that every vertex in Z1 has exactly one neighbor in M and at most one vertex
in M has no neighbor in Z1 (else there is a chair or a homogeneous set) and thus G is
a thin spider. ⋄
By symmetry, we have:
Claim 9.3.3 If G contains gem then it is a thick spider.
Finally, if G is gem- and co-gem-free then, by Lemma 9.3, it is diamond- and codiamond-free. In this case, the following is easy to see by using Lemma 9.1 (cf. e.g.
[98]):
Claim 9.3.4 If P4 -sparse split graph G is diamond- and co-diamond-free then it is a
P4 or bull.
This shows one direction of Theorem 9.5. For the converse direction, assume that all
prime subgraphs of G are spiders. We have to show that G is (P5 , house, chair, co −
chair, P, co − P, C5 )-free. Obviously, spiders have this property, and looking at homogeneous sets of G, it is easy to see that none of the subgraphs P5 , house, chair, co −
chair, P, co − P, C5 can have some vertices in a homogeneous set H and some others
out of H. This shows Theorem 9.5.
2
Various structural and algorithmic consequences for P4 -sparse graphs are given in [289,
290, 292, 275, 276].
Other examples for simple prime graphs are induced paths and cycles as well as their
complements, matched co-bipartite graphs and their complements as well as bipartite
chain graphs and their complements. Various classes can be characterized in terms of
such prime graphs or at least have such prime graphs.
Various results on generalizations of P4 -sparse graphs were published. Fouquet and Giakoumakis [216] studied so-called semi-P4 -sparse graphs which are the (P5 ,co-chair,coP5 )-free graphs:
Theorem 9.6 ([216]) If G is a prime (P5 ,co-P5,co-chair)-free graph then G is either
a bipartite chain graph or a spider or C5 .
Theorem 9.7 ([100])
(i) If G is a prime (chair,co-P ,house)-free graph then G is either a co-matched bipartite graph or a spider or an induced path or cycle.
(ii) If G is a prime (chair,co-chair,P ,co-P )-free graph then G or G is either a spider
or an induced path or cycle or has at most 9 vertices.
These two classes also deserve the name semi-P4-sparse graphs.
Here are some other examples (which do not generalize P4 -sparse graphs):
99
Theorem 9.8 ([77])
(i) If G is a prime chair-free bipartite graph then G is a co-matched bipartite graph
or an induced path or cycle.
(ii) If G is a prime (bull,chair)-free graph with stability number at least 4 then G is
co-matched bipartite or an induced path or cycle.
(iii) If G is a prime (bull,chair,co-chair)-free graph then G or G is co-matched bipartite
or an induced path or cycle.
The situation becomes a bit more complicated for (P5 ,diamond)-free and, more generally, for (P5 ,gem)-free graphs (cf. [56, 84]) but there are helpful structural properties
of prime graphs of these classes. In the chapter on clique-width, various consequences
are mentioned for all these classes; in particular, simple prime graphs lead to bounded
clique-width.
Here are two examples of subclasses of chordal graphs with simple prime graphs:
9.3.2
Structure of co-chair-free chordal graphs
The structure of gem-free chordal graphs is well known (see Theorem 8.10). Now, by
switching to the complement graphs, assume that G is a co-chordal graph containing
an induced co-gem; we can assume that G is prime. Moreover, since G is co-chordal,
it is 2K2 - and C5 -free, and, as the complement of a co-chair-free chordal graph, it is
chair-free. Recall Lemma 9.4.
Eight of the nine graphs in Fig. 2 of Lemma 9.4 contain 2K2 , C5 or chair. The
only graph not containing any of them is the net. From now on assume that prime
(2K2 , C5 ,chair)-free graph G contains net, say N with vertices a1 , a2 , a3 , b1 , b2 , b3 such
that a1 , a2 , a3 is a stable set (the end-vertices of N), b1 , b2 , b3 is a clique (the mid-vertices
of N), and the only edges between a1 , a2 , a3 and b1 , b2 , b3 are ai bi ∈ E, i = 1, 2, 3. We
claim:
Lemma 9.6 If prime (2K2 , C5 ,chair)-free graph G = (V, E) contains an induced subgraph isomorphic to net then G is a thin spider.
Proof. The proof is done by a number of claims. Let N be a net as described above
with vertex set V (N). We classify R = V \ V (N) with respect to the number of
neighbors of a vertex x ∈ R in N. Vertex x ∈ R is an i-vertex with respect to N if
x∈
/ V (N) and x has exactly i neighbors in N, 0 ≤ i ≤ 6. Let Mi denote the set of
i-vertices of N, 1 ≤ i ≤ 5, and let Z denote the set of 0-vertices and U denote the set
of 6-vertices. Since G is 2K2 -free, Z is a stable vertex set. We claim that N has no
1-vertex, no 2-vertex, and no 5-vertex:
Claim 9.3.5 M1 ∪ M2 ∪ M5 = ∅.
100
Proof of Claim 9.3.5:
First assume that x is a 1-vertex of N: If xa1 ∈ E then xa1 , a2 b2 is a 2K2 . If xb1 ∈ E
then x, a1 , b1 , b2 , a2 induce a chair.
Now assume that x is a 2-vertex of N: If xa1 ∈ E and xa2 ∈ E then x, a1 , b1 , b2 , a2
induce a C5 . If xa1 ∈ E and xb1 ∈ E then xa1 , b2 a2 is a 2K2 . If xa1 ∈ E and xb2 ∈ E
then a1 , x, b2 , b3 , a2 induce a chair. If xb1 ∈ E and xb2 ∈ E then a1 , b1 , x, b3 , a3 induce
a chair.
Finally assume that x is a 5-vertex of N: If xa1 ∈
/ E then a1 , b1 , x, a2 , a3 induce a chair.
If xb1 ∈
/ E then b1 , a1 , x, a2 , a3 induce a chair. ⋄
As a next step we prove that 3- and 4-vertices have a restricted type of neighborhood
in N:
Claim 9.3.6 If x ∈ M3 then x sees either exactly one end-vertex ai and the two
opposite mid-vertices bj , bk , j 6= i, k 6= i or x sees all three mid-vertices of N.
Proof of Claim 9.3.6: Assume that 3-vertex x is nonadjacent to at least one of the
mid-vertices. If x sees all three end-vertices a1 , a2 , a3 then x, a1 , b1 , b2 , a2 induce a C5 .
If x sees two end-vertices, say, xa1 ∈ E and xa2 ∈ E and one mid-vertex then for
xb3 ∈ E, x, a1 , b1 , b2 , a2 induce a C5 , and for xb1 ∈ E, xa1 , b2 b3 is a 2K2 . If x sees
one end-vertex, say, xa1 ∈ E and two mid-vertices then for xb1 ∈ E and xb2 ∈ E,
the vertices a1 , x, b2 , b3 , a2 induce a chair, and analogously, for xb1 ∈ E and xb3 ∈ E,
the vertices a1 , x, b3 , b2 , a3 induce a chair. Thus, only the case xa1 ∈ E, xb2 ∈ E and
xb3 ∈ E is admissible, and symmetrically for the cases xa2 ∈ E, xa3 ∈ E, respectively.
⋄
Let Mid3 denote the set of 3-vertices adjacent to all three mid-vertices of N. The
situation is similar for 4-vertices:
Claim 9.3.7 If x ∈ M4 then it sees exactly one end-vertex and all mid-vertices.
Proof of Claim 9.3.7: Let x be a 4-vertex of N. Obviously, a 4-vertex sees at least one of
the end-vertices, say xa1 ∈ E. If x sees all three end-vertices a1 , a2 , a3 and, say xb1 ∈ E
then x, a2 , b2 , b3 , a3 induce a C5 . If x sees two end-vertices, say xa1 ∈ E, xa2 ∈ E then
if xb3 ∈
/ E then a1 x, b3 a3 is a 2K2 , and if xb3 ∈ E then a3 , b3 , x, a1 , a2 induce a chair.
Thus, Claim 9.3.7 holds. ⋄
1
Claim 9.3.8 U (M
3 ∪ M4 ).
Proof of Claim 9.3.8: Assume that there are u ∈ U and x ∈ M3 with ux ∈
/ E. If
x ∈ Mid3 then x, b1 , u, a2 , a3 induce a chair. If xa1 ∈ E then x, a1 , u, a2, a3 induce a
chair.
Now assume that there are u ∈ U and x ∈ M4 with ux ∈
/ E. Then again, if xa1 ∈ E
then x, a1 , u, a2 , a3 induce a chair. ⋄
101
c
Let Z1 denote the set of 0-vertices having a neighbor in M3 ∪ M4 , and let Z0 := Z \ Z1 .
0
Claim 9.3.9 Z1 ((M
3 ∪ M4 ) \ Mid3 ).
Proof of Claim 9.3.9: Assume that z ∈ Z1 and x ∈ (M3 ∪ M4 ) \ Mid3 with xa1 ∈ E
and xb3 ∈ E with xz ∈ E. Then z, x, a1 , b3 , a3 induce a chair. ⋄
Thus, the only possible neighbors of Z1 vertices in M3 ∪ M4 are the vertices in Mid3 .
1 1.
Claim 9.3.10 U Z
Proof of Claim 9.3.10: Assume that u ∈ U and z ∈ Z1 with uz ∈
/ E. Let x ∈ Mid3 be
a neighbor of z. Then z, x, u, a2 , a3 induce a chair. ⋄
Now, V (N) ∪ M3 ∪ M4 ∪ Z1 is a module: Every vertex u ∈ U sees all of them, and
every vertex z ∈ Z0 misses all of them. Since G is prime, V (N) ∪ M3 ∪ M4 ∪ Z1 must
be a trivial module, i.e., V = V (N) ∪ M3 ∪ M4 ∪ Z1 and thus U ∪ Z0 = ∅.
We still have to show that G[V (N) ∪ M3 ∪ M4 ∪ Z1 ] is a thin spider.
Let Mi′ := (M3 ∪ M4 ) ∩ N(ai ), i = 1, 2, 3.
1 j′ .
Claim 9.3.11 For i 6= j, Mi′ M
Proof of Claim 9.3.11: If for x ∈ M1′ and y ∈ M2′ xy ∈
/ E then xa1 , ya2 is a 2K2 . ⋄
1
Claim 9.3.12 For every i = 1, 2, 3, Mi′ Mid
3.
Proof of Claim 9.3.12: If for x ∈ M1′ and y ∈ Mid3 xy ∈
/ E then a1 , x, b2 , y, a2 is a
chair. ⋄
Thus, for every i = 1, 2, 3, Mi′ ∪ {bi } is a module, i.e., Mi′ = ∅. It remains to show:
Claim 9.3.13 Mid3 is a clique.
Proof of Claim 9.3.13: Assume that Mid3 is not a clique; let Q denote the co-connected
component in G[Mid3 ] containing a non-edge. Then there are vertices x, y ∈ Mid3 with
xy ∈
/ E and there is a vertex z ∈ Z1 distinguishing x and y, say xz ∈ E and yz ∈
/ E.
Now z, x, b3 , y, a3 induce a chair. ⋄
So far, we have shown that G is a chair-free split graph. In Lemma 7 of [101] we show
that a prime chair-free split graph is a spider; actually, in our case, it is a thin spider
since the spider contains net. This finally shows Lemma 9.6.
2
102
Theorem 9.9 If prime graph G is co-chair-free and chordal then G is either a block
graph or a thick spider.
Proof. Let G be a prime co-chair-free and chordal graph. If G is gem-free then,
by Lemma 9.3, G is diamond-free and thus G is a block graph. Now assume that G
contains gem. Then by Lemmas 9.4 and 9.6, G is a thin spider and thus, G is a thick
spider.
2
9.3.3
Structure of bull-free chordal graphs
Recall Lemma 9.1: If prime graph G contains an induced K3 then G contains bull,
house or double-gem. Note that double-gem is the complement graph of A where A
has six vertices, four of which induce a C4 and two others are leaves attached to two
successive vertices in the C4 , say A has vertices v1 , v2 , v3 , v4 , v5 , v6 such that v2 , v3 , v4 , v5
induce a C4 with edges v2 v3 , v3 v4 , v4 v5 , v5 v2 , v1 is adjacent to v2 and v6 is adjacent to
v5 . Let v1 and v6 be the end-vertices of A and the other ones the mid-vertices of A.
Note that double-gem is the complement graph of A where A has six vertices, four of
which induce a C4 and two others are leaves attached to two successive vertices in the
C4 , say A has vertices v1 , v2 , v3 , v4 , v5 , v6 such that v2 , v3 , v4 , v5 induce a C4 with edges
v2 v3 , v3 v4 , v4 v5 , v5 v2 , v1 is adjacent to v2 and v6 is adjacent to v5 . Let v1 and v6 be the
end-vertices of A and the other ones the mid-vertices of A.
We first describe the structure of bull-free co-chordal graphs containing an induced
subgraph isomorphic to A; recall that co-chordal graphs are 2K2 -free and C5 -free.
Slightly more generally, we claim:
Lemma 9.7 If prime (2K2 ,bull)-free graph G contains an induced subgraph isomorphic
to A then G is a bipartite chain graph.
Proof. Let G be a prime (2K2 , C5 ,bull)-free graph containing an induced subgraph A
with vertices and edges as above. We analyze the properties of i-vertices of A in G. As
before, let Mi denote the set of i-vertices of A, 1 ≤ i ≤ 5, and let Z denote the set of
0-vertices and U denote the set of 6-vertices. Since G is 2K2 -free, Z is a stable vertex
set. We claim that A has no 5-vertex:
Claim 9.3.14 M5 = ∅.
Proof of Claim 9.3.14: If x is a 5-vertex with xv1 ∈
/ E then v1 , v2 , v3 , v6 , x induce a bull.
If x is a 5-vertex with xv2 ∈
/ E then v2 , v3 , v4 , v6 , x induce a bull. If x is a 5-vertex with
/ E then v1 , v3 , v4 , v5 , x induce a bull. The other cases are symmetric. ⋄
xv3 ∈
Obviously, the following holds since G is 2K2 -free:
Claim 9.3.15 If x ∈ M1 then xv2 ∈ E or xv5 ∈ E, and M1 is a stable set.
103
Let N24 (N35 , respectively) denote the set of common neighbors of v2 and v4 (of v3 and
v5 , respectively) in M2 .
Claim 9.3.16 If x ∈ M2 then x ∈ N24 ∪ N35 . Moreover, N24 and N35 are stable sets.
Proof of Claim 9.3.16: Since G is 2K2 -free, xv1 ∈ E implies xv5 ∈ E or xv6 ∈ E,
xv5 ∈ E or xv4 ∈ E, and xv3 ∈ E or xv4 ∈ E. Now let x ∈ M2 . Thus, if x is adjacent
to the two end-vertices then xv1 , v3 v4 is a 2K2 in G. If x is adjacent to exactly one
end-vertex, say xv1 ∈ E (and xv6 ∈
/ E) then by the previous argument, xv5 ∈ E but
now xv1 , v3 v4 is a 2K2 in G. Thus, xv1 ∈
/ E and by symmetry, xv6 ∈
/ E. If xv3 ∈ E
then, since xv3 , v5 v6 is not a 2K2 in G, we have xv5 ∈ E, and similarly for xv4 ∈ E.
Finally, if xv3 ∈
/ E and xv4 ∈
/ E then, since x ∈ M2 , xv2 ∈ E and xv5 ∈ E and now
v1 , v2 , v5 , v6 , x induce a bull.
If for x, y ∈ N24 xy ∈ E then xy, v5 v6 is a 2K2 , and analogously for N35 . Thus, N24
and N35 are stable sets. ⋄
Let N135 (N246 , respectively) denote the set of common neighbors of v1 , v3 and v5 (of
v2 , v4 and v6 , respectively) in M3 .
Claim 9.3.17 If x ∈ M3 then x ∈ N135 ∪ N246 .
Proof of Claim 9.3.17: Let x ∈ M3 . The arguments are similar as in the proof of Claim
9.3.16. Assume first that x is adjacent to the two end-vertices. Then, since xv1 , v3 v4
is not a 2K2 , we have xv3 ∈ E or xv4 ∈ E, say xv3 ∈ E but now xv1 , v4 v5 is a 2K2 .
If x is adjacent to exactly one end-vertex, say xv1 ∈ E (and xv6 ∈
/ E) then again, by
the 2K2 argument, xv5 ∈ E. Moreover, xv3 ∈ E or xv4 ∈ E must hold but if xv4 ∈ E
then v1 , v4 , v5 , v6 , x induce a bull. Thus, x ∈ N135 , and similarly for xv6 ∈ E. Finally,
if xv1 ∈
/ E and xv6 ∈
/ E then for xv2 ∈ E and xv5 ∈ E, v1 , v2 , v5 , v6 , x induce a bull,
and thus either xv2 ∈
/ E or xv5 ∈
/ E but then we have a 2K2 . ⋄
Let N1235 (N2456 , respectively) denote the set of common neighbors of v1 , v2 , v3 and v5
(of v2 , v4 , v5 and v6 , respectively) in M4 .
Claim 9.3.18 If x ∈ M4 then x ∈ N1235 ∪ N2456 .
Proof of Claim 9.3.18: Let x ∈ M4 . The arguments are similar as in the proof of
Claims 9.3.16 and 9.3.17. Assume first that x is adjacent to the two end-vertices.
Then, since xv1 , v3 v4 is not a 2K2 , we have xv3 ∈ E or xv4 ∈ E, say xv3 ∈ E but
now for xv2 ∈ E, v2 , v3 , v4 , v6 , x induce a bull, and symmetrically for xv5 ∈ E, and for
xv4 ∈ E, v2 , v3 , v4 , v6 , x induce a bull.
If x is adjacent to exactly one end-vertex, say xv1 ∈ E (and xv6 ∈
/ E) then again,
by the 2K2 argument, xv5 ∈ E. Moreover, xv3 ∈ E or xv4 ∈ E but for xv4 ∈ E,
v1 , v4 , v5 , v6 , x induce a bull. Thus xv3 ∈ E and xv4 ∈
/ E, and now x ∈ N1235 .
Finally, if xv1 ∈
/ E and xv6 ∈
/ E then v1 , v2 , v5 , v6 , x induce a bull. ⋄
Next we show:
104
Claim 9.3.19 N1235 = N2456 = ∅ and N135 ∪ {v2 } (N246 ∪ {v5 }, respectively) is a stable
set.
Proof of Claim 9.3.19: Suppose to the contrary that N1235 6= ∅; let u ∈ N1235 . We first
claim that every connected component in N135 ∪ N1235 ∪ {v2 } (in N246 ∪ N2456 ∪ {v5 },
respectively) is a module; if it is one-elementary, this is trivially the case. Thus, suppose
that it has at least two vertices. Without loss of generality, let Qu be the connected
component in G[N135 ∪ N1235 ∪ {v2 }] containing v2 and u. Since G is prime, Qu is not
a homogeneous set in G, and thus, there is a pair x, y ∈ Qu with xy ∈ E and there is a
vertex z ∈
/ Qu such that zx ∈ E and zy ∈
/ E. Since zx ∈ E, we have z ∈
/ N135 ∪ N1235 ∪
{v2 }. Let Ax (Ay , respectively) denote the induced subgraph G[{v1 , v3 , v4 , v5 , v6 } ∪{x}]
(which is isomorphic to A) (G[{v1 , v3 , v4 , v5 , v6 } ∪ {y}], respectively). If z is an i-vertex
for Ay then z is an (i+ 1)-vertex for Ax . Thus, by Claim 9.3.14, z is not a 4- or 5-vertex
for Ay .
Since z, x, y, v3 , v4 do not induce a bull, we have zv3 ∈ E or zv4 ∈ E, since z, x, y, v5 , v4
do not induce a bull, we have zv5 ∈ E or zv4 ∈ E, and since z, x, y, v5 , v6 do not induce
a bull, we have zv5 ∈ E or zv6 ∈ E.
Case 1. zv5 ∈ E.
Then, as above, also zv3 ∈ E or zv4 ∈ E and z is a 2- or 3-vertex for Ay , i.e., by Claims
9.3.16 and 9.3.17, z ∈ N35 ∪ N135 for Ay but for z ∈ N35 , x, z, v1 , v3 , v5 induce a bull,
and by z ∈
/ Qu we have z ∈
/ N135 .
Case 2. zv5 ∈
/ E.
Since z, x, y, v4 , v5 is not a bull, zv4 ∈ E. Likewise, since z, x, y, v5 , v6 is not a bull,
zv6 ∈ E. Since v3 y, zv6 is not a 2K2 , zv3 ∈ E but now y, z, v3 , v4 , v6 induce a bull contradiction which shows N1235 = ∅.
The arguments for showing N2456 = ∅ are symmetric. Thus, N1235 = N2456 = ∅ holds,
i.e, M4 = ∅.
Now, with very similar arguments, we can show that N135 ∪ {v2 } (N246 ∪ {v5 },
respectively) is a stable set. This shows Claim 9.3.19. ⋄
Let M := M1 ∪ M2 ∪ M3 , let Z1 be the set of vertices in Z having a neighbor in M
and let Z0 := Z \ Z1 . Since G is 2K2 -free, obviously, if z ∈ Z1 then for any neighbor
x ∈ M of z, x ∈ M3 . Thus:
0
Claim 9.3.20 Z1 (M
1 ∪ M2 ).
Let Z1′ be the set of vertices in Z having a neighbor in N135 and let Z1′′ be the set of
vertices in Z having a neighbor in N246 .
Claim 9.3.21 Z1′ ∩ Z1′′ = ∅.
Proof of Claim 9.3.21: Suppose to the contrary that there are z ∈ Z1 , x ∈ N135 and
y ∈ N246 such that zx ∈ E and zy ∈ E. Then if xy ∈
/ E, v1 , x, y, v6 induce a 2K2 and
if xy ∈ E, v1 , x, y, z, v6 induce a bull - contradiction. ⋄
105
1
Claim 9.3.22 U (M
∪ Z1 ∪ {v1 , . . . , v6 }).
1
Proof of Claim 9.3.22: By definition, U {v
1 , . . . , v6 }. Let u ∈ U. First suppose that
x ∈ M and ux ∈
/ E. If x ∈ N2 ∪ N24 then x, v1 , v2 , v6 , u induce a bull, and similarly for
x ∈ N5 ∪ N35 . If x ∈ N246 then x, v3 , v5 , v6 , u induce a bull, and similarly for x ∈ N135 .
Now suppose that z ∈ Z1 and uz ∈
/ E. Then z has a neighbor x ∈ M3 , say x ∈ N135 .
If uz ∈
/ E then z, x, v1 , v4 , u induce a bull. Thus, Claim 9.3.22 holds. ⋄
0
Claim 9.3.23 Z0 (M
∪ Z1 ∪ {v1 , . . . , v6 }).
0
Proof of Claim 9.3.23: By definition, Z0 (M
∪ {v1 , . . . , v6 }), and recall that Z is a
0
stable set since G is 2K2 -free. Thus, Z0 Z1 holds. ⋄
Claim 9.3.24 U ∪ Z0 = ∅.
Proof of Claim 9.3.24: By Claims 9.3.22 and 9.3.23, Z1 ∪M ∪{v1 , . . . , v6 } is a module in
G. Since G is prime, this must be a trivial module, i.e., V (G) = Z1 ∪ M ∪ {v1 , . . . , v6 }
which shows Claim 9.3.24. ⋄
Claim 9.3.25 G is bipartite.
Proof of Claim 9.3.25: Let X := {v1 , v3 , v5 } ∪ N2 ∪ N24 ∪ N246 ∪ Z1′ and let Y :=
{v2 , v4 , v6 } ∪ N5 ∪ N35 ∪ N135 ∪ Z1′′ . The previous claims imply that X (Y , respectively)
is a stable vertex set, and thus, G = (X ∪ Y, E) is bipartite. ⋄
Since G is 2K2 -free, Lemma 9.7 is shown. (Note that we did not use C5 in the arguments.)
2
Theorem 9.10 If prime graph G is bull-free and chordal then G is either a tree or a
co-bipartite chain graph.
Proof. Let G be a prime bull-free and chordal graph. If G is triangle-free then clearly
G is a tree. Now assume that G contains a triangle. Then by Lemma 9.1, G contains
double-gem, i.e., G contains A and G is (2K2 , C5 ,bull)-free. By Lemma 9.7, G is a
bipartite chain graph which shows Theorem 9.10.
2
Note that Theorem 9.10 also follows from a result in [383]; the graphs called k-web in
[383] are the prime co-bipartite chain graphs.
106
9.4
Split Decomposition
A lot of research has been done in generalizing, refining and modifying modular decomposition. Split (or join) decomposition was introduced and studied by Cunningham
[162]. A graph is split decomposable if its vertex set has a partition into A1 , A2 and
B1 , B2 such that A = A1 ∪ A2 , B = B1 ∪ B2 , and the set of all edges between A and B
1 1 . The decomposition is discussed in detail in the monograph [414]
forms a join A1 B
by Spinrad, mentioning the linear time algorithm for split decomposition by Dahlhaus
[167]. A simplified linear-time algorithm for split decomposition is given in [135].
The class of graphs such that every induced subgraph on at least four vertices is
decomposable by the join decomposition is of particular interest. It turns out that these
are exactly the distance-hereditary graphs which are the central topic of Chapter 8.
9.5
Homogeneous Decomposition
A natural extension of modular decomposition is the interesting concept of homogeneous decomposition introduced by Jamison and Olariu [291] based on P4 ’s of a graph.
The importance of this concept also comes from Reed’s theorem on the P4 -structure
(which was conjectured by Chvátal):
Theorem 9.11 ([393]) If graphs G and G′ have the same P4 -structure then G is
perfect if and only if G′ is perfect.
In homogeneous decomposition, a third operation is added which is a combination of
join and co-join. As for modular decomposition, it yields a unique decomposition tree
for arbitrary graphs. The approach is based on generalized connectedness - the pconnectedness introduced by B. Jamison and Olariu in [291]. The paper [17] by Babel
and Olariu gives a survey on this concept and corresponding results.
Definition 9.1 ([291]) Let G = (V, E) be a graph.
For a partition V1 , V2 of V , a P4 is a crossing P4 between V1 and V2 if it contains
vertices from V1 and V2 .
G is p-connected if every partition V1 , V2 of V has a crossing P4 .
The p-connected components of G are the maximal induced subgraphs which are
p-connected.
Obviously, the p-connected components are connected subgraphs of G and G. Furthermore, each graph G has a unique partition into p-connected components and so-called
weak vertices, which are not contained in any P4 .
107
Definition 9.2 ([291]) A p-connected graph G = (V, E) is separable if V can be
partitioned into nonempty V1 , V2 in such a way that each crossing P4 has its midpoints
in V1 and its endpoints in V2 . Then V1 , V2 is a separation of G.
The complement of a separable p-connected graph is clearly also separable.
Theorem 9.12 ([291]) Every separable p-connected graph has a unique separation.
Furthermore, every vertex belongs to a crossing P4 with respect to the separation.
Theorem 9.13 ([291]) A p-connected graph G is separable if and only if its characteristic graph G∗ is a split graph.
The p-connectedness structure theorem for graphs is similar to Theorem 9.4 on substitution decomposition, and Theorem 9.5 on P4 -sparse graphs:
Theorem 9.14 ([291]) For an arbitrary graph G exactly one of the following conditions holds:
(i) G is not connected.
(ii) G is not connected.
(iii) There is a unique proper separable p-connected component H of G with a partition
(H1 , H2 ) such that every vertex outside H sees all vertices in H1 and misses all
vertices in H2 .
(iv) G is p-connected.
Theorem 9.14 leads to the homogeneous (also called primeval) decomposition tree as
follows. The labels of the interior nodes of the decomposition tree correspond to the
first three cases of Theorem 9.14, while the leaves are the p-connected components
and weak vertices.
Baumann [27] showed that the homogeneous decomposition tree can be constructed
in linear time.
The following notion of a p-chain leads to a characterization of p-connected graphs:
Definition 9.3 ([12, 16]) Let G be a graph. A p-chain connecting x and y is
a sequence of distinct vertices (v1 , . . . , vk ) such that x = v1 , y = vk and for all
i ∈ {1, . . . , k − 3} the set Xi = {vi , vi+1 , vi+2 , vi+3 } induces a P4 in G.
The simplest examples for p-chains are given by the P4 s of a Pk , k ≥ 5. There are,
however, more interesting examples. The importance of the notion of p-chains becomes
more clear in the following
108
Theorem 9.15 ([12, 16]) A graph is p-connected if and only if every pair of vertices
in the graph is connected by a p-chain.
Babel and Olariu define a generalization of cographs which has similarities to trees:
Definition 9.4 ([15, 12]) Let G be a graph.
G is a p-cycle if every vertex of G is contained in at least two P4 s and G is
minimal with this property i.e. every proper induced subgraph of G has a vertex
occurring in at most one P4 .
G is a p-tree if G is p-connected and contains no p-cycle.
G is a p-forest if G contains no p-cycle.
Note that there are simple trees which are p-cycles. However, it is worth noting that a
p-cycle with at least 8 vertices is P4 -isomorphic to a chordless cycle [12].
Definition 9.5 (Babel [12]) Let G be a p-connected graph.
A vertex v is a p-articulation vertex in G if G − v is not p-connected.
v is a p-end-vertex if v is contained in exactly one P4 .
Theorem 9.16 (Babel [12]) Let G be a graph. The following conditions are equivalent:
(i) G is a p-tree;
(ii) G is p-connected and every p-connected induced subgraph H of G contains at least
one p-end-vertex;
(iii) G is p-connected, contains no proper induced headless spider and has exactly n−3
P4 s;
(iv) G contains no induced p-cycle and has exactly n − 3 P4 s;
(v) G is p-connected, contains no proper induced headless spider and each vertex of a
p-connected induced subgraph H of G is either a p-end-vertex or a p-articulation
vertex in H;
(vi) G contains no proper induced headless spider and each pair of vertices is connected
either by a unique nontrivial p-chain or by trivial p-chains only.
Babel [12] shows that recognition and isomorphism of p-trees can be done in linear
time.
The P4 -structure of some fundamental graph classes is investigated in various papers;
see e.g. [90] for the P4 -structure of trees and [14] for the P4 -structure of bipartite
graphs.
109
9.6
Clique Separator Decomposition
An important amount of work has been done on clique separator decomposition introduced by Whitesides [435] (see also Tarjan [420]): A clique separator (or clique cutset)
is a clique whose removal increases the number of components of the graph. In other
words: A clique separator of a graph G is a separator of G which induces a complete
subgraph in G.
For a chordal graph G which is not a complete graph, and a simplicial vertex v in G,
obviously N(v) is a clique separator of G. Clique separator decomposition of a graph is
generalizing chordal graphs by repeatedly choosing a clique separator in G until there
is no longer a clique separator in the resulting subgraphs; such subgraphs are called
atoms of G. Thus, decomposition by clique separators consists in repeatedly copying
a clique separator into the components it defines; the final set of subgraphs obtained,
which are then devoid of clique separators, are the atoms of G.
More formally: Given a graph G with a clique separator S, where C is a component of
G − S, graph G is decomposed into subgraphs G(C ∪ S) (called an S-component) and
G − C, which are in turn decomposed until none of the obtained subgraphs contain a
clique separator; these final subgraphs are the atoms.
A peripheral atom A of G has only one clique separator S, i.e., V (A) = B ∪ S such
that B is a maximal (connected) set of vertices which belong to only one atom in G,
and S is a clique separator of G. The decomposition step using S produces in this case
atom A = G(B ∪ S) and the rest graph G − B.
The following is well known: In a chordal graph, the atoms are the maximal complete
subgraphs. Subsequently, atoms which are a complete subgraph will be called clique
atoms.
Theorem 9.17 A graph is chordal if and only if its atoms are complete graphs.
Clique separator decomposition is hole- and C4 - preserving (see [42] for a comprehensive
survey on this decomposition).
For computing clique separator decomposition of a given graph G, an algorithm for
finding a clique separator of G devised by Whitesides [435] can be used; see [420] for a
discussion of the running time O(nm). As Tarjan explains, the time required depends
on the number of atoms. Gavril [229] gave an upper bound for this which leads to
time bound O(n3 m) for finding a decomposition. Tarjan [420] gives a faster algorithm
by using minimal fill-in and perfect elimination orderings based on [381, 404]; his time
bound is O(nm) and his decomposition produces at most n − 1 atoms:
Theorem 9.18 ([420]) A clique separator decomposition of a given graph can be determined in time O(nm), and it leads to at most n − 1 atoms of G.
Note that clique separator decomposition trees of arbitrary graphs are not uniquely
determined. This was improved by Leimer [322] in the following way: The decomposition by minimal clique separators decomposes a graph (in a unique fashion) into atoms
110
called MP-subgraphs in [322], which are characterized as maximal induced connected
subgraphs containing no clique separator.
Minimal triangulation of a given graph consists in adding an inclusion-minimal set of
edges to obtain a chordal graph. This is a topic of self-standing interest and has given
rise to many recent papers. The time bound for this problem, originally O(nm) [404],
was recently lowered to O(n2.69 ) time [314] and even O(nα log n) = o(n2.376 ) [271]. A
special issue of Discrete Mathematics (see [37]) is published on the subject, containing
a survey on minimal triangulation [270].
Clique separator decomposition has a number of algorithmic applications described in
[420], among them efficiently solving the Maximum Weight Independent Set (MWIS)
problem on a graph class whenever it is efficiently solvable on the atoms of the class.
This refers to the weight modification approach described in the algorithm of Frank
for the same problem on chordal graphs - see Theorem 3.6. More exactly, the MWIS
problem is recursively solved as follows (we slightly modify Tarjan’s approach in [420]):
We first describe a contraction step for a peripheral atom G[A∪C] with clique separator
C in graph G = (V, E). Let B := V \ (A ∪ C), and let w denote the vertex weight
function in G. For a vertex set U ⊆ V , let w(U) := Σu∈U w(u). For an induced
subgraph H of G, let αw (H) denote the maximum weight of an independent vertex set
in H.
Contraction Step. For peripheral atom G[A∪C] of G, construct GA = (VA , EA )
by creating a new vertex vA representing A such that {vA } ∪ C is a clique (and
thus, VA := {vA } ∪ C ∪ B and EA is the set of edges {va x | x ∈ C} plus
all edges in G[C ∪ B]), and set w ′ (vA ) := αw (G[A]) and for every x ∈ C, set
w ′ (x) := w(x) + αw (G[A \ N(x)]) as the modified weight function w ′ in GA .
Correctness: We claim that αw (G) = αw′ (GA ): Obviously, any independent set in G
contains at most one vertex from C, i.e., for a maximum-weight independent set I in
G, I ∩ C ≤ 1 holds.
Case 1. I ∩ C = 1, say, x ∈ I ∩ C:
Then I = {x}∪I ′ ∪I ′′ for an independent set I ′ in G[A\N(x)] with maximum w-weight
in G[A \ N(x)] and I ′′ an independent set in G[B \ N(x)] with maximum w-weight in
G[B \ N(x)]; the part of G[A \ N(x)] is represented in the new weight w ′(x). Thus,
w(I) = w(x) + w(I ′ ) + w(I ′′ ) and w ′ ({x} ∪ I ′′ ) = w ′ (x) + w ′ (I ′′ ) = w(x) + w(I ′ ) + w(I ′′ ),
i.e., αw (G) = αw′ (GA ).
Case 2. I ∩ C = 0:
In this case, I = I ′ ∪ I ′′ for a maximum w-weight independent set I ′ in G[A] (i.e.,
w(I ′) = αw (G[A]) = w ′ (vA ) and a maximum w-weight independent set I ′′ in G[B].
Thus, w(I) = w(I ′) + w(I ′′) and w ′({vA } ∪ I ′′ ) = w ′ (vA ) + w ′ (I ′′ ) = w(I ′) + w(I ′′ ), i.e.,
αw (G) = αw′ (GA ) holds in this case as well.
Time bound: The running time for reducing the atom G[A∪C] to a simplicial vertex vA
with weight w ′ (vA ) := αw (G[A]) mainly depends on the running time for determining
111
diamond
dart
gem
paraglider
co-C6
Figure 17: diamond, dart, gem, paraglider, and co-C6.
αw (G[A ∪ C]). In [420], it is mentioned that one has to solve O(n) independent set
problems per atom, for a total of O(n2 ) subproblems.
Elimination Step. Because vA is simplicial in GA , we can eliminate vA and
continue working on the rest graph, by applying Claim 3.1.1 from the MWIS
algorithm of Frank.
Now, by successively contracting peripheral atoms G[A ∪ C] and eliminating the simplicial vertex vA generated in this way, we finally obtain a p.e.o. of the reduced graph
and, according to Frank’s algorithm, an optimal solution of the MWIS problem in
polynomial time if MWIS is solvable in polynomial time for the atoms of the graph
class C.
Various examples of graph classes having atoms of simple structure (generalizing or
modifying the special case of chordal graphs) were studied: Tarjan [420] is mentioning
two examples, namely clique separable graphs defined and studied by Gavril [229] and
EPT graphs studied by Golumbic and Jamison [239, 240].
In [74], a subclass of hole-free graphs, namely hole- and paraglider-free graphs, is
characterized by the structure of their atoms. Among others, this is motivated by a
result of Alekseev [6] showing that atoms of (P5 ,paraglider)-free graphs are 3K2 -free
which implies polynomial time for Maximum Weight Independent Set on this class.
For P5 -free graphs, the complexity of the MWIS problem was open for a long time;
meanwhile, it has been shown by Lokshtanov et al. [337] that it is polynomially solvable
for P5 -free graphs. For hole-free graphs, the complexity of the MWIS problem is open.
Recall that block graphs are the diamond-free chordal graphs; their blocks are complete
subgraphs, and clique separators are one-elementary called cut vertices or articulation
points. A natural way generalizing block graphs is to consider diamond-free hole-free
graphs (i.e., C4 is admitted). In [39], the atoms of these graphs are described as follows:
Theorem 9.19 ([39]) Let A be an atom of a hole- and diamond-free graph. Then A
is
(i) a complete subgraph, or
(ii) matched co-bipartite, or
(iii) chordal bipartite.
112
Theorem 9.19 describes the structure of a subclass of hole- and paraglider-free graphs.
For these, a characterization via atoms is given:
Theorem 9.20 ([74]) A graph G is hole- and paraglider-free if and only if every atom
of G is either
(i) a complete multipartite graph, or
(ii) the join of a matched co-bipartite graph and a (possibly empty) clique, or
(iii) the join of a chordal bipartite and a (possibly empty) clique.
In [39], a LexBFS property concerning clique separators is given:
Recall the characterization of LexBFS orderings by the 4-point condition in Theorem
3.3. We say that vertex d results from a < b < c.
Theorem 9.21 Let G = (V, E) be a graph, let α be a LexBFS ordering of G, and S a
clique separator of G. Then for any component C of G(V − S), the sub-ordering β of
α on V − C is a LexBFS ordering of G(V − C).
Proof. Suppose that for a component D of G(V −S), sub-ordering β of α on G(V −D)
fails to be a LexBFS ordering of G(V − D): There are vertices a < b < c in β but no
d > c fulfills condition (1) in β; since α is a LexBFS ordering of G, by condition (1),
such a vertex d must exist in α, and thus, d ∈ D holds. Since bd ∈ E, b ∈ S follows;
likewise, since bc ∈
/ E, c ∈
/ S follows. Therefore, in G, c and d lie in two different
components C and D of G(V − S), say c ∈ C and d ∈ D.
We claim that the repeated application of condition (1) leads to infinitely many new
resulting vertices (which is a contradiction since G is finite):
Since b ∈ S, c ∈ C − S, d ∈ D − S and bd ∈ E, cd ∈
/ E, we can apply condition (1) to
b < c < d; let e be a resulting vertex; ce ∈ E and be ∈
/ E. Thus, e ∈
/ S which means
that e ∈ C − S. Moreover, de ∈
/ E. Thus, we can apply condition (1) to c < d < e; let
f be a resulting vertex and so on. Now, as long as the new vertices are not in S, the
procedure continues until a resulting vertex, say w, is in S. Then bw ∈ E, and since
w resulted from, say u < v < x, uw ∈
/ E and (1) can be applied to b < u < w; let y
be a resulting vertex. Note that u and v are in different components u ∈ C and v ∈ D
or vice versa. Since by ∈
/ E, y is in C − S if u ∈ C − S or in D − S if u ∈ D − S,
respectively. Now the process continues with u < v < y since v and y are in different
components until again a resulting vertex is in S but then the same argument as above
can be applied. This shows Theorem 9.21.
2
In [75], clique separator decomposition and modular decomposition are combined, however, it has to be defined in a more accurate way as done in [73] since Corollary 9 of
[75] is promising too much. The correct version is:
113
Whenever MWIS is solvable in polynomial time for the atoms of prime graphs
of a hereditary graph class C then MWIS is solvable in polynomial time for the
class C itself [73].
In [73], this approach is applied to hole- and co-chair-free graphs and corrects and
extends the results of [72] (which were based on Corollary 9). A similar example is the
class of apple-free graphs generalizing chordal graphs, claw-free graphs and cographs;
in [97], the approach of [75] was applied but also in this case, it can be replaced by
investigating atoms of prime graphs and leads to a polynomial time algorithm for
MWIS on these graphs.
114
10
10.1
Treewidth and Clique-Width of Graphs
Treewidth
Treewidth of a graph measures its tree-likeness. Treewidth of trees has value one,
and if the treewidth of a graph class is bounded by a constant, this has important
consequences for the efficient solution of many problems on the class. Treewidth was
introduced by Bertelé and Brioschi [43] under the name of dimension, was rediscovered
by Halin [261] and later by Robertson and Seymour in their famous graph minor
project (see e.g. [399, 400, 401, 402]). Treewidth is one of the most important concepts
of algorithmic graph theory. It also came up in the framework of partial k-trees which
have many applications (see e.g. [8]). Good surveys are given by Bodlaender [49] and
by Kloks [306].
We first define k-trees recursively:
Definition 10.1 Let k ≥ 1 be an integer. The following graphs are k-trees:
(i) Any clique Kk with k vertices is a k-tree.
(ii) Let G = (V, E) be a k-tree, let x ∈
/ V be a new vertex and let C ⊆ V be a clique
′
with k vertices. Then also G = (V ∪ {x}, E ∪ {ux | u ∈ C}) is a k-tree.
(iii) There are no other k-trees.
It is easy to see that for k = 1, the k-trees are exactly the trees, and for any k, k-trees
are chordal with maximum clique size k + 1 if the graph is not a clique. More exactly,
all maximal cliques have size k + 1 in this case. See [403] for simple characterizations
of k-trees.
Definition 10.2 Graph G′ = (V, E ′ ) is a partial k-tree if there is a k-tree G = (V, E)
with E ′ ⊆ E.
Obviously, every graph with n vertices is a partial n-tree, and every k-tree is a partial
k-tree. The following parameter is of tremendous importance for the efficient solution
of algorithmic problems on graphs.
Definition 10.3 The treewidth tw(G) of a given graph G is the minimum value k for
which G is a partial k-tree.
Theorem 10.1 ([8]) Determining the treewidth of a graph is NP-complete.
Treewidth was defined in a different way by Robertson and Seymour (see e.g. [399,
400, 401, 402]) via tree decompositions of graphs:
115
Definition 10.4 A tree decomposition of a graph G = (V, E) is a pair D = (S, T )
with the following properties:
(i) S = {Vi | i ∈ I} is a finite collection of subsets of vertices (sometimes called
bags).
(ii) T = (I, F ) is a tree with one node for each subset from S.
S
(iii) i∈I Vi = V .
(iv) For all edges (v, w) ∈ E, there is a subset (i.e., a bag) Vi ∈ S such that both v
and w are contained in Vi .
(v) For each vertex x ∈ V , the set of tree nodes {i | x ∈ Vi } forms a subtree of T .
Condition (v) corresponds to the join tree condition of α-acyclic hypergraphs and to
the clique tree condition of chordal graphs. Thus, a graph is chordal if and only if it
has a tree decomposition into cliques.
The width of a tree decomposition is the maximum bag size minus one. It is not hard
to see that the following holds (see e.g. [306]):
Lemma 10.1 The treewidth of a graph equals the minimum width over all of its tree
decompositions.
The fundamental importance of treewidth for algorithmic applications is twofold: First
of all, many problems can be solved by dynamic programming in a bottom-up way along
a tree decomposition (or equivalently, an embedding into a k-tree) of the graph, and
the running time is “quite good” for “small” k. [9, 10, 35, 48] give many examples for
this approach. Secondly, there is a deep relationship to Monadic Second Order Logic
described in various papers by Courcelle [157] (and in many other papers of this author;
see also Bodlaender’s tourist guide [49]). Roughly speaking, the following holds:
Whenever a problem Π is expressible in Monadic Second Order Logic and C is a graph
class of bounded treewidth (with given tree decomposition for each input graph) then
problem Π can be efficiently solved on every input graph from C.
As an example, consider 3-colorability of a graph (which is well known to be NPcomplete):
∃W1 ⊆ V ∃W2 ⊆ V ∃W3 ⊆ V ∀v ∈ V (v ∈ W1 ∨ v ∈ W2 ∨ v ∈ W3 ) ∧ ∀v ∈ V ∀w ∈
V (vw ∈ E ⇒ (¬(v ∈ W1 ∧w ∈ W1 )∧¬(v ∈ W2 ∧w ∈ W2 )∧¬(v ∈ W3 ∧w ∈ W3 )))
The detour via logic, however, leads to astronomically large constant factors in the
running time of such algorithms. Therefore it is of crucial importance to have a tree
decomposition of the input graph with very small width. We know already (see Theorem 10.1) that the problem of determining treewidth is NP-complete.
116
Theorem 10.2 (Bodlaender [50]) For each integer k ≥ 1 there is a linear time
algorithm which for given graph G either determines that tw(G) > k holds or otherwise
finds a tree decomposition with width k.
Some classes of graphs (such as cactus graphs, series-parallel graphs, Halin graphs,
outerplanar graphs etc.) have bounded treewidth. On the other hand, complete graphs
Kn have treewidth n, and n × n square grids have treewidth n and thus, planar graphs
have unbounded treewidth. See [49] for more information.
Thorup [423] gives important examples of small treewidth in computer science applications.
Closely related parameters are pathwidth and bandwidth (see [51]); based on interval
graphs, the pathwidth of a graph can be defined analogously as treewidth based on
chordal graphs; the tree decomposition for determining pathwidth has linear structure.
Bandwidth is based on proper interval graphs instead of interval graphs. Determining
the bandwidth of caterpillars is NP-complete [365]. See also [297].
Another closely related graph parameter called tree-length is proposed by Dourisboure
and Gavoille [?]. It measures how close a graph is to being chordal. The tree-length of
G is defined using tree decompositions of G (see Definition 10.4). Graphs of tree-length
k are the graphs that have a tree decomposition where the distance in G between any
pair of vertices that appear in the same bag of the tree decomposition is at most k.
We discuss this and related parameters in Section 13.
10.2
Clique-Width of Graphs
The notion of clique-width of a graph, defined by Courcelle, Engelfriet and Rozenberg
(in the context of graph grammars) in [158], is another fundamental example of a
width parameter on graphs which leads to efficient algorithms for problems expressible
in some kind of Monadic Second Order Logic.
More formally, the clique-width cw(G) of a graph G is defined as the minimum number
of different integer labels which allow to generate graph G by using the following four
kinds of operations on vertex-labeled graphs:
(i) Creation of a new vertex v labeled by integer ℓ (denoted by ℓ(v)).
(ii) Disjoint union of two (vertex-labeled and vertex-disjoint) graphs G1 , G2 (i.e.,
co-join, denoted by G1 ⊕ G2 ).
(iii) Join between the set of all vertices with label i and the set of all vertices with
label j for i 6= j (i.e., all edges between the two sets are added, denoted by ηi,j ).
(iv) Relabeling of all vertices of label i by label j (denoted by ρi→j ).
A k-expression for a graph G of clique-width k describes the recursive generation of
G by repeatedly applying these operations (i) − (iv) using at most k different labels.
Here is a 3-expression for generating C5 = (v1 , v2 , v3 , v4 , v5 , v1 ) with three labels 1,2,3:
117
η3,1 (ρ3→2 (η2,3 (η1,2 (1(v1 ) ⊕ 2(v2 )) ⊕ η1,3 (3(v3 ) ⊕ 1(v4 )))) ⊕ 3(v5 ))
Arbitrarily long induced cycles (paths, respectively) can be generated with four labels
(three labels, respectively). For example, the following recursive procedure generates
Pk = (v1 , v2 , . . . , vk ) with three labels:
(1) Create v1 with label 1. Create v2 with label 2. Carry out join η1,2 . Create v3
with label 3. Carry out join η2,3 . Relabel v2 by label 1, and relabel v3 by label 2.
(2) Repeatedly, carry out i-th step, i ≥ 4: Create vertex vi with label 3, and join η2,3
(remember: vi−1 has label 2). Relabel vi−1 by label 1, and relabel vi by label 2.
Obviously, any graph with n vertices can be generated using n labels (for each vertex
a specific one). Thus cw(G) ≤ n if G has n vertices. By Definition 8.3, it is not hard
to see that the class of cographs is exactly the class of graphs having clique-width at
most 2, and a 2-expression can be found in linear time along the cotree of a cograph:
Proposition 10.1 ([160]) For graph G, cw(G) ≤ 2 if and only if G is a cograph.
Determining the clique-width of a given graph is a complicated task even for relatively
small graphs. Solving a long-standing open problem, in [212], Fellows, Rosamond,
Rotics, and Szeider show that determining clique-width is NP-complete:
Theorem 10.3 ([212]) Given a graph G and an integer k, deciding whether cw(G) ≤
k is NP-complete.
The recognition problem for graphs of clique-width at most 3 is solvable in polynomial
time [147]. For any fixed k ≥ 4, the problem of recognizing all graphs with clique-width
at most k in polynomial time is open.
Clique-width is more powerful than treewidth in the sense that if a class of graphs has
bounded treewidth then it also has bounded clique-width but not vice versa [160] the clique-width of cliques of arbitrary size is 2 whereas their treewidth is unbounded.
In particular, an upper bound for the clique-width of a graph is obtained from its
treewidth as follows:
Theorem 10.4 (Corneil, Rotics [156]) For any graph G, cw(G) ≤ 3 · 2tw(G)−1 .
Another relationship between treewidth and clique-width is given by:
Theorem 10.5 (Gurski, Wanke [255]) For a graph class, treewidth is bounded if
and only if for its line graphs, clique-width is bounded.
118
Similarly as for treewidth, the concept of clique-width of a graph has attracted much
attention due to the fact that there is a similarly close connection to Monadic Second
Order Logic: In [159], Courcelle, Makowsky and Rotics have shown that every graph
problem definable in LinMSOL(τ1 ) (a variant of Monadic Second Order Logic using
quantifiers on vertex sets but not on edge sets) is solvable in linear time on graphs with
bounded clique-width if a k-expression describing the input graph is given.
The problems Maximum Weight Stable Set, Maximum Weight Clique, k-Coloring for
fixed k, Steiner Tree and Domination are examples of LinMSOL(τ1 ) definable problems
whereas Coloring and Hamiltonian Circuit are not.
Theorem 10.6 (Courcelle, Makowsky, Rotics [159]) Let C be a class of graphs
of clique-width at most k such that there is an O(f (|E|, |V |)) algorithm, which for
each graph G in C, constructs a k-expression defining it. Then for every LinMSOL(τ1 )
problem on C, there is an algorithm solving this problem in time O(f (|E|, |V |)).
Moreover, for some other problems which are not expressible in this way, there are
polynomial time algorithms for classes of bounded clique-width [202, 308, 231].
In [159, 160], various fundamental clique-width properties are shown. Clique-width is
closely related to modular decomposition as the following proposition shows:
Proposition 10.2 ([159, 160]) Let G be a graph.
(i) cw(G) = max{cw(H) | H is a prime subgraph of G}.
(ii) cw(G) ≤ 2cw(G).
See [296] for further properties concerning operations such as local complementation,
bipartite complement and others.
For various classes of graphs, clique-width is unbounded. The arguments differ; the
reason might be that there are too many graphs in the class - see e.g. [296]: For
a graph class X, let Xn denote the number of labeled graphs with n vertices in X.
In [408], hereditary graph classes are classified into five levels: constant, polynomial,
exponential, factorial and super-factorial. The first three have bounded clique-width
[296]. A class of graphs X is said to be factorial if Xn satisfies the inequalities nc1 n ≤
Xn ≤ nc2 n for some constants c1 and c2 .
If there is no constant c such that Xn ≤ ncn then X is said to be super-factorial.
Examples of factorial classes are planar graphs, permutation graphs, and line graphs.
Bipartite, co-bipartite and split graphs are super-factorial graph classes since the num2
ber of n-vertex graphs in these classes is at least 2n /4 . The following theorem shows
that super-factorial classes have unbounded clique-width.
Theorem 10.7 ([52]) If a class of graphs is super-factorial, then its clique-width is
unbounded.
119
Thus, for bipartite graphs, co-bipartite graphs and split graphs, clique-width is unbounded. For various other classes, a certain kind of grid structure leads to unbounded
clique-width. Examples are square grids, some other kinds of grids [353], unit interval
graphs [244] (recall the grid structure of these graphs described in Theorem 3.21) and
bipartite permutation graphs [96] (with similar grid structure).
In [88], it is shown that gem- and co-gem-free graphs have bounded clique-width. This
means the following:
Theorem 10.8 ([88]) The clique-width of graphs G such that for every vertex of G,
its neighborhood and its non-neighborhood in G induces a cograph in G, is bounded.
It is easy to see that the clique-width of thin spiders is at most 4. Thus, a simple
consequence of Theorem 9.5 and Proposition 10.2 is that the clique-width of P4 -sparse
graphs is bounded. Many other examples are given in [66] by combining forbidden
induced subgraphs which are one-vertex extensions of a P4 or by combining forbidden
induced subgraphs with four vertices [69].
It is easy to see that the clique-width of trees is at most 3: Trees can be recursively
generated by adding a new root r to a collection of (smaller) trees Ti with distinguished
root vertex ri , i = 1, . . . , k: We can inductively assume that r1 , . . . , rk have label 2 and
is the only vertex with label 2 in Ti (all other vertices in Ti have label 1), and the new
root r has label 3. Then carry out η2,3 and relabel by ρ3→2 .
The fact that the clique-width of distance-hereditary graphs is at most 3 (which, at
first glance, does not seem to be surprising but the proof is quite technical) is based
on pruning sequences (see Theorem 8.6):
Theorem 10.9 (Golumbic, Rotics [244]) The clique-width of distance-hereditary
graphs is at most 3, and corresponding 3-expressions can be constructed in linear time.
Various other classes of bounded and unbounded clique-width are described in [66, 69,
88, 89, 319, 353] and many other papers. See [296] for recent results on graph classes
of bounded (and unbounded) clique-width. In [311], it is shown that split permutation
graphs have unbounded clique-width; recall that G is a split permutation graph if and
only if G and G is an interval graph (see Theorem 3.18).
10.2.1
Clique-Width of H-Free Bipartite Graphs
Recall that prime P5 -free bipartite graphs are chain graphs whose clique-width is at
most 3. Lozin [343] has shown that the clique-width of P6 -free bipartite graphs is
bounded. In [69], it follows from the proof of Theorem 11 that the clique-width of
P8 -free bipartite graphs is unbounded. In [345] it is shown that the clique-width of
P7 -free bipartite graphs is unbounded. Finally, in [164], a complete dichotomy result
for classifying the clique-width of H-free bipartite graphs is given.
120
10.2.2
Clique-Width of H-Free Chordal Graphs
Recently, [62] describes the attempt for a complete dichotomy result for classifying the
clique-width of H-free chordal graphs is given which is done up to two stubborn open
cases; see also [165].
As a consequence of Theorem 10.9 and 8.10, the clique-width of gem-free chordal graphs
is at most 3. It is a natural question to ask what happens for other graphs H. In [89],
it is shown that also the clique-width of co-gem-free chordal graphs is bounded. In
the same paper, it is erroneously claimed that gem and co-gem are the only 1-vertex
extensions H of P4 for which the clique-width of H-free chordal graphs is bounded.
Subsequently, we show that also for H = bull and H = co − chair, the clique-width of
H-free chordal graphs is bounded.
By Theorem 9.9, we obtain:
Corollary 10.1 The clique-width of co-chair-free chordal graphs is bounded.
By Theorem 9.10, we obtain:
Corollary 10.2 The clique-width of bull-free chordal graphs is bounded.
Clearly, for H = P5 , H = co − P , H = P , H = C5 and H = co − P5 , the clique-width of
H-free chordal graphs is unbounded since it is unbounded for split graphs. Moreover,
for H = chair, the clique-width of H-free chordal graphs is unbounded since it is
unbounded for unit interval graphs and unit interval graphs are claw-free.
In [165], the clique-width of H-free chordal graphs is studied for any induced subgraph
H; there are some open cases such as when H is a 4-clique Q with two leaves attached
to different vertices of Q. In [69], it is shown that the clique-width of 4P1 -free chordal
graphs is unbounded by showing for the complement graphs:
Theorem 10.10 ([69]) K4 -free co-chordal graphs have unbounded clique-width.
For the proof of Theorem 10.10, the following family of grids is analyzed. For each
integer n ≥ 1, let Gn be the graph with vertex set A ∪ B ∪ C and edge set E1 ∪ E2 ∪ E3
defined as follows:
Consider a (n + 1) × (n + 1) square grid with vertex vij in row i and column j,
i, j ∈ {0, . . . , n}. Omit the vertex v00 , take the vertices of the 0-th column as the
A-vertices ai = vi0 , take the vertices of the 0-th row as the B-vertices bi = v0i and take
the other vertices as the C-vertices cij = vij , 1 ≤ i, j ≤ n.
The edges between A, B and C are defined as follows.
• A and B induce a complete bipartite graph; E1 is the set of all edges ai bj ,
1 ≤ i, j ≤ n.
121
• Every vertex ai ∈ A is adjacent to every j-th row, 1 ≤ j ≤ i; E2 is the set of all
edges ai vjℓ , 1 ≤ j ≤ i, 1 ≤ ℓ, i ≤ n.
• Every vertex bi ∈ B is adjacent to every j-th column, 1 ≤ j ≤ i; E3 is the set of
all edges bi vℓj , 1 ≤ j ≤ i, 1 ≤ ℓ, i ≤ n.
Thus, A, B, and C are stable sets, |A| = |B| = n, and |C| = n2 such that A∪B induces
a complete bipartite graph, and A ∪ C as well as B ∪ C induce a bipartite chain graph.
See Figure 18 for the graph G3 .
A−vertices:
B−vertices:
C−vertices:
Figure 18: The graph G3
Observation 10.1 For each n ≥ 1, Gn is a K4 -free co-chordal graph.
Proof. Since V = A ∪ B ∪ C is a partition into three stable sets, Gn is clearly K4 -free.
In particular, Gn is Ck -free for all k ≥ 8. Moreover, it is a matter of routine to check
that Gn is also Ck -free for k ∈ {4, 5, 6, 7}, i.e., Gn is co-chordal.
2
Lemma 10.2 For each n ≥ 1, cwd(Gn ) ≥ n.
This may be surprising at first glance, since Gn consists of a pair of bipartite chain
graphs G[A∪C], G[B ∪C] with the property that A and B form a join, and is thus quite
close to bipartite chain graphs whose clique-width is at most 3 but Gn has unbounded
clique-width.
Proof of Lemma 10.2. Let τ be a k-expression constructing Gn and let T := tree(τ )
be the corresponding construction tree of Gn . Let x be a ⊕ node in T with children
y, z such that
(i) neither the induced subgraph of Gn defined by tree(y, τ ) nor the induced subgraph of Gn defined by tree(z, τ ) contains a full row or a full column of Gn ,
but
(ii) the induced subgraph of Gn defined by tree(x, τ ) contains a full row or a full
column of Gn .
122
We color all vertices of Gn which are leaves of tree(y, τ ) with color red and all vertices
of Gn which are leaves of tree(z, τ ) with color blue. Color the remaining vertices of Gn
with color white.
We say that a vertex v of Gn is nonwhite if v is red or blue. A row (column) is red
(blue or nonwhite, respectively) when all vertices in this row (column) are red (blue or
nonwhite, respectively).
Note that by (i),
there is no blue (red, respectively) row in Gn at x, and
there is no blue (red, respectively) column in Gn at x.
(6)
Moreover, by (ii),
there is a nonwhite row or a nonwhite column in Gn at x.
(7)
Furthermore,
edges between red vertices and blue vertices (if any) are not yet constructed at x,
(8)
white vertices (if any) are not yet constructed at x.
(9)
and
Claim 10.2.1 Let u, u′ be two nonwhite vertices in Gn . Suppose there is a vertex v
adjacent to u but nonadjacent to u′ . If color(v) 6= color(u) then label(u) 6= label(u′ ) at
x.
Proof of Claim 10.2.1: As color(v) 6= color(u) and u is a nonwhite vertex, the edge
between v and u is not yet constructed at x by (8), (9). Suppose label(u) = label(u′ )
at x. Then label(u) = label(u′ ) at every ancestor vertex of x in tree(τ ). Thus, when
creating the edge between v and u with ηlabel(v),label(u) at an ancestor of x, the edge
between v and u′ would also be created, in contradiction to the fact that v and u′ are
nonadjacent. This shows Claim 10.2.1.
Now, suppose without loss of generality that, by (7), there is a nonwhite column in
Gn at x. Thus, every row i ≥ 1 contains a nonwhite vertex. Let ji be the smallest
index such that viji is a nonwhite vertex in row i, i ≥ 1. Recall that none of the rows
is completely red and none of them is completely blue.
Claim 10.2.2 The nonwhite vertices viji , 1 ≤ i ≤ n, have pairwise different labels at
x.
Proof of Claim 10.2.2: For i 6= i′ , consider two such distinct nonwhite vertices ui := viji
and ui′ := vi′ ji′ . We have to show that label(ui ) 6= label(ui′ ) at x. Note that i, i′ ≥ 1.
Case 1. ui ∈ A, ui′ ∈ C (i.e., ji = 0, ji′ ≥ 1).
By (6), there is a vertex v ∈ C in row i with color(v) 6= color(ui ). By definition of Gn ,
123
v is adjacent to ui and nonadjacent to ui′ (recall that C is stable). By Claim 10.2.1,
label(ui ) 6= label(ui′ ).
Case 2. ui , ui′ ∈ A (i.e., ji = ji′ = 0).
Without loss of generality, let i < i′ . By (6), there is a vertex v ∈ C in row i′ with
color(v) 6= color(ui′ ). By definition of Gn , v is adjacent to ui′ and nonadjacent to ui
because i < i′ . Hence by Claim 10.2.1, label(ui ) 6= label(ui′ ).
Case 3. ui , ui′ ∈ C (i.e., ji , ji′ ≥ 1).
Without loss of generality, let i < i′ . By the choice of ji , ai is a white vertex. By
definition of Gn and the assumption i < i′ , ai is adjacent to ui and nonadjacent to ui′ .
Hence by Claim 10.2.1, label(ui ) 6= label(ui′ ). This shows Claim 10.2.2.
Now Claim 10.2.2 implies Lemma 10.2.
2
Observation 10.1 and Lemma 10.2 together show that K4 -free co-chordal graphs have
unbounded clique-width which finishes the proof of Theorem 10.10.
2
10.2.3
Clique-Width of (H1 , . . . , Hk )-Free Graphs
In [163], it is shown that the clique-width of H-free graphs is bounded if and only if
H is an induced subgraph of P4 . In [69], a complete classification of all graph classes
defined by forbidden induced subgraphs of at most four vertices is given. In [163], this
is extended to forbidden induced subgraphs of any size studying the clique-width of
(H1 , H2 )-free graphs (but the classification is not complete). We give some examples:
In [83], it is shown that the clique-width of (P6 , K3 )-free graphs is bounded while in
[163], it is shown that the clique-width of (P6 ,diamond)-free graphs is unbounded.
10.2.4
Power-Bounded Clique-Width of Graph Classes
In [53], various graph classes are investigated with respect to power-bounded (and
power-unbounded) clique-width which is defined as follows: A graph class C has powerbounded clique-width if there is an integer k such that the k-th powers of graphs in C
have bounded clique-width. If no such k exists, C has power-unbounded clique-width;
examples of such classes are:
Theorem 10.11 ([53]) Bipartite permutation graphs and unit interval graphs have
power-unbounded clique-width.
Another example of results in [53] is:
Theorem 10.12 ([53]) For every graph H, the class of H-free graphs has powerbounded clique-width if and only if H is a linear forest.
124
10.3
NLC-Width and Rank-Width of Graphs
The notion of NLC-width introduced by Wanke [433] is closely related to clique-width.
The NLC-width of a graph is not greater than its clique-width, and the clique-width
of a graph is twice its NLC-width [294]. Computing the NLC-width of a graph is NPcomplete [254]. The graphs of NLC-width 1 are the cographs, and the class of graphs
of NLC-width at most 2 can be recognized in polynomial time [328]. Similarly as for
clique-width (with k ≥ 4), recognition of NLC-width at most k is open for k ≥ 3.
Oum and Seymour [384, 385, 386] investigated the important concept of rank-width
and its relationship to clique-width, treewidth and branchwidth. In particular, Oum
showed:
Theorem 10.13 ([384, 385])
(i) A class of graphs has bounded rank-width if and only if it has bounded cliquewidth.
(ii) The rank-width of a graph G is 1 if and only if G is distance hereditary.
125
11
Leaf Powers
One of the fundamental problems in computational biology is to reconstruct the evolutionary history of a set of species, based on quantitative biological data. Typically,
the evolutionary history is modeled by an evolutionary tree called the phylogeny which
is a tree whose leaves are labeled by species and in which each internal node represents
a speciation event whereby an ancestral species gives rise to two or more child species
[136, 330].
Motivated by this background, Nishimura, Ragde and Thilikos [380] defined the following notions: A tree T is a k-leaf root of a finite undirected graph G = (V, E) if V
is the set of leaves of T and for any two vertices x, y ∈ V , xy ∈ E if and only if the
distance of x and y in T is at most k. Graph G is a k-leaf power if it has a k-leaf root;
it is a leaf power if it is a k-leaf power for some k ≥ 2. (Note that Lin, Kearney and
Jiang [330] defined the notion of k-th phylogenetic power and k-root phylogeny of G in
a similar way with the condition that internal tree nodes have degree at least three.)
Obviously, a graph is a 2-leaf power if and only if it is the disjoint union of cliques.
Meanwhile characterizations of 3-leaf powers and of 4-leaf powers as well as linear
time recognition algorithms for 3-, 4- and 5-leaf powers are known while for k ≥ 6,
characterizing k-leaf powers as well as characterizing leaf powers is a challenging open
problem.
The aim of this chapter is to give a survey on some recent results on leaf powers.
11.1
Basic notions and results
Replacing vertex v in graph G by graph H (or substituting H into v) results in the
graph obtained from (G − v) ∪ H by adding all edges between vertices in NG (v) and
vertices in H. In particular, H might be a clique; thus, for instance 3-leaf powers are
exactly the graphs resulting from substituting cliques into the vertices of a tree (see
Theorem 11.12).
Since inclusion-maximal clique modules are exactly the equivalence classes of relation
R, where vRw if and only if N[v] = N[w], the subsequent Proposition 11.1 is a wellknown fact; see e.g. [394] by Roberts on indifference graphs.
Proposition 11.1 The inclusion-maximal clique modules of a graph are pairwise disjoint.
In [330], Lin, Kearney and Jiang call the inclusion-maximal clique modules of G =
(V, E) critical cliques of G, and they define the critical clique graph CC(G) of G as the
graph having the critical cliques of G as its nodes, and two distinct nodes Q and Q′
are adjacent in CC(G) if there are vertices x ∈ Q and y ∈ Q′ such that xy ∈ E. Note
that CC(G) has no true twins.
126
A linear time algorithm for constructing the critical clique graph CC(G) for a given
chordal graph G was already claimed in [330]; such an algorithm is given e.g. in [40, 354]
where the maximal clique modules of a (not necessarily chordal) graph are constructed
in linear time.
11.2
Basic facts on leaf powers
This section deals with the relationship of leaf powers, fixed tolerance NeST graphs
and strongly chordal graphs. Moreover, it characterizes unit interval graphs as those
leaf powers having caterpillar leaf roots and discusses the leaf rank of leaf powers which
is the smallest k such that a leaf power G has a k-leaf root.
11.2.1
Leaf powers are strongly chordal
In [168, 349, 392], it is shown that the class of strongly chordal graphs is closed under
powers:
Theorem 11.1 ([168, 349, 392]) If G is strongly chordal then for every k ≥ 1, Gk
is strongly chordal.
This is a simple consequence of Theorem 3.8 (all odd powers of a chordal graph are
chordal), Corollary 5.1 (the square of a dually chordal graph is chordal, and thus all
even powers are chordal) and the fact that strongly chordal graphs are dually chordal
- see Corollary 7.1.
Let T be a k-leaf root of k-leaf power G. Then, by definition, G is isomorphic to the
subgraph of T k induced by the leaves of T , i.e., G is an induced subgraph of the power
of a tree. Since trees are strongly chordal and induced subgraphs of strongly chordal
graphs are strongly chordal, this implies by Theorem 11.1 (see also [300]):
Proposition 11.2 For every k ≥ 2, k-leaf powers are strongly chordal.
This strengthens the observation that k-leaf powers are chordal (see e.g. [185, 380, 391]).
Proposition 11.3 Let k ≥ 2.
(i) Every induced subgraph of a k-leaf power is a k-leaf power.
(ii) A graph is a k-leaf power if and only if each of its connected components is a
k-leaf power.
127
11.2.2
Leaf powers and fixed tolerance NeST graphs are the same class
It is natural to ask whether the containment of leaf powers in the class of strongly
chordal graphs is strict. It turns out that this question can be answered by showing
that a graph is a leaf power if and only if it is a so-called fixed tolerance NeST graph
investigated by Bibelnieks and Dearing [45] and by Hayward, Kearney and Malton
[268, 269]. The abbreviation NeST stands for neighborhood subtree tolerance graphs.
We are not going to define NeST graphs; instead we will simply use the following fact
from [268, 269]:
Theorem 11.2 A graph G = (V, E) is a fixed tolerance NeST graph if and only if
there is a positive constant k > 0 and an undirected weighted tree T = (N, A, ω) with
V ⊆ N and positive real weights ω : A → R+ on the edges such that for all u, v ∈ V
uv ∈ E ⇐⇒ dT (u, v) ≤ k.
Note that in the original formulation of Theorem 11.2, instead of using positive weights
ω in trees, in [45, 268, 419], trees are embedded into the plane (and dT (u, v) is given by
the distance between u and v in the plane embedding of T ); in fact, these two models
are equivalent. In [45], based on [110], it is shown:
Theorem 11.3 Fixed tolerance NeST graphs are strongly chordal, and there are
strongly chordal graphs which are no fixed tolerance NeST graph.
In [81], we show:
Theorem 11.4 A graph is a leaf power if and only if it is a fixed tolerance NeST
graph.
This has several consequences. First of all, as a corollary of Theorems 11.3 and 11.4, it
shows that leaf powers are a proper subclass of strongly chordal graphs: See Figure 19
for an example of such a graph.
Corollary 11.1 There are strongly chordal graphs which are no leaf power.
Since interval graphs are fixed tolerance NeST graphs [269], Theorem 11.4 implies that
interval graphs are leaf powers which was shown in a direct way in [80]. We strengthen
this inclusion in [81] by showing:
Theorem 11.5 Every rooted directed path graph is a leaf power but not vice versa.
Recall Theorem 8.10 about ptolemaic graphs. Using the recursive generation of
distance-hereditary graphs via pendant vertex, true twin and false twin operation (see
Theorem 8.6), Dahlhaus showed:
Theorem 11.6 (Dahlhaus [166]) Every ptolemaic graph is a rooted directed path
graph.
Corollary 11.2 Ptolemaic graphs are leaf powers.
128
Figure 19: A strongly chordal graph which is not a leaf power.
11.2.3
Leaf rank and unit interval graphs
The leaf rank of a leaf power G is the smallest k such that G is a k-leaf power. It seems
to be hard to determine the leaf rank of a given leaf power. We know the leaf rank of
leaf powers only in some very restricted cases - see Lemma 11.1 for an example.
For the next result we implicitly use the concept of clique-width (see section 10.2).
Recall that Golumbic and Rotics [244] showed that the clique-width of unit interval
graphs is unbounded.
By a result of Todinca [424] (see also [254]), for every fixed k, the class of k-leaf powers
has bounded clique-width since k-th powers of a graph class of bounded clique-width
(such as trees) have bounded clique-width. Assuming that unit interval graphs have
bounded leaf rank would imply bounded clique-width for unit interval graphs, which
is a contradiction. Thus we obtain:
Proposition 11.4 The leaf rank of unit interval graphs is unbounded.
Thus also interval graphs and rooted directed path graphs have unbounded leaf rank.
Another consequence of the fact that unit interval graphs are leaf powers is:
Corollary 11.3 Leaf powers have unbounded clique-width.
A caterpillar T is a tree consisting of a (chordless) path (the backbone of T ) and some
leaves attached to the backbone. In [80], we gave a new characterization of unit interval
graphs in terms of caterpillar leaf roots as well as in terms of induced subgraphs of
powers of paths (see Theorem 3.21):
Theorem 11.7 For a graph G, the following conditions are equivalent:
(i) G has a leaf root which is a caterpillar.
(ii) G is an induced subgraph of the power of some induced path.
129
(iii) G is a unit interval graph.
k−2
The following powers P2k−3
of induced paths P2k−3 with 2k −3 vertices are unit interval
graphs for which the leaf rank can be determined:
Lemma 11.1 ([106, 431]) For all k and k ′ with 2 ≤ k ′ < k, the (k − 2)-th power
k−2
P2k−3
of the induced path P2k−3 with 2k − 3 vertices is a k-leaf power but no k ′ -leaf
power.
k−2
This means that the leaf rank of P2k−3
is k. For some upper and lower bounds on the
leaf rank of interval graphs and of ptolemaic graphs see [80, 81].
11.3
The inclusion structure of k-leaf power classes
For k ≥ 2, let L(k) denote the class of k-leaf powers. Let G be a k-leaf power and let
tree T be a k-leaf root of G. It is easy to see that L(k) ⊂ L(k + 2) (by subdividing
edges of T containing a leaf), and L(k) ⊂ L(2k − 2) as well as L(k) ⊂ L(2k − 1) (by
subdividing edges of T not containing a leaf). Thus, for instance L(2) ⊂ L(3) ⊂ L(4)
and L(4) ⊂ L(6) as well as L(4) ⊂ L(7).
For a while, however, it was unknown whether L(k) ⊂ L(k + 1) holds for every k ≥ 2.
The first (negative) result in this direction was the proof that L(4) 6⊂ L(5) given by
Fellows et al. [211]. In [106] we generalized this result to all k:
Theorem 11.8 For every k ≥ 4, L(k) 6⊂ L(k + 1).
In [431], we give a complete description of containments between k-leaf powers and
k ′ -leaf powers:
Theorem 11.9 For all k ≥ 2 and ℓ ≥ 1, the following holds:
(i) L(k + ℓ) 6⊂ L(k).
(ii) L(k) 6⊂ L(k + ℓ) if and only if ℓ is odd and ℓ ≤ k − 3.
For example, L(6) ⊂ L(8) as well as L(6) ⊂ L(11) but L(6) 6⊂ L(7) and L(6) 6⊂ L(9).
Theorem 11.9 implies:
Corollary 11.4 For all k and k ′ with 2 ≤ k < k ′ , the inclusion L(k) ⊂ L(k ′ ) holds if
and only if for any G ∈ L(k), a k ′ -leaf root of G can be obtained by correspondingly
subdividing edges of a k-leaf root of G.
Theorem 11.9 (i) follows by Lemma 11.1. The proof of Theorem 11.9 (ii) is long and
very technical. Among others, it makes use of Buneman’s Four-Point Condition – see
Theorem 8.13 and the remark before this theorem for the metric similarity between
trees and block graphs in terms of Buneman’s four-point condition in section 8.5.2.
130
11.4
Characterizations of 3-leaf powers, 4-leaf powers and
distance-hereditary 5-leaf powers
In this section we present various characterizations of 3-leaf powers as well as of 4leaf powers but also a new characterization of squares of trees and some other results
concerning distance-hereditary 5-leaf powers.
11.4.1
Basic leaf powers and substitution of cliques
The following notion from [94] simplifies many arguments.
A k-leaf root T of a k-leaf power G is basic if to every internal node of T at most one
leaf is attached. A k-leaf power is basic if it has a basic k-leaf root. An internal node
in a tree is called invisible if no leaf is attached to it.
Observation 11.1
(i) Every induced subgraph of a basic k-leaf power is a basic k-leaf power.
(ii) A k-leaf power without true twins is a basic k-leaf power.
Note that the other direction of Observation 11.1 (ii) does not hold; cf. the comments
after Proposition 11.6.
Proposition 11.5 For every graph G, and for every k ≥ 2, G is a k-leaf power if and
only if G is obtained from a basic k-leaf power G′ by substituting nonempty cliques into
the vertices of G′ .
Proposition 11.6 Let k ≥ 3. Then for every graph G,
(i) G is a basic k-leaf power having a basic k-leaf root without invisible vertices if
and only if G is the (k − 2)-th power of some tree.
(ii) G is a k-leaf power if and only if G is obtained from the (k − 2)-th power of some
tree by substituting the vertices by (possibly empty) cliques.
Note that (i) in Proposition 11.6 gives examples for basic k-leaf powers that may
contain true twins. For example, the clique with k vertices minus an edge Kk− (which
is the (k − 2)-th power of the induced path Pk ) is a basic k-leaf power, but contains
true twins provided k ≥ 4.
Corollary 11.5 Let k ≥ 3 and G be a graph. Then G is a basic k-leaf power if and
only if G is an induced subgraph of the (k − 2)-th power of some tree.
131
11.4.2
Characterizations of 3-leaf powers
Recall that a graph is a 2-leaf power if and only if it is the disjoint union of cliques.
Thus, in the hierarchy of k-leaf powers, the class of 3-leaf powers represents the first
nontrivial class. By attaching leaves to all vertices of a tree, it is easy to see that the
following holds:
Observation 11.2 Every tree (forest, respectively) is a 3-leaf power, and a 3-leaf root
of it can be determined in linear time.
Theorem 11.10 ([85, 391]) A connected graph G is a 3-leaf power if and only if G
is the result of substituting cliques into the vertices of a suitable tree.
Proof. First assume that the connected graph G = (V, E) is a 3-leaf power, and let T
be a 3-leaf root for G with leaf set V . Let V = V1 ∪ . . . ∪ Vk be a partition of V with
respect to common parent nodes in T , i.e., Vi is the (nonempty) set of T -leaves in V
having the same parent node ti in T .
Claim 11.4.1 If for all i ∈ {1, . . . , k}, |Vi | = 1 then G is a tree.
Proof. Assume to the contrary that G contains a cycle C. Since G, as a 3-leaf power,
is chordal, C is a C3 , say with vertices a, b, c. Let x′ be the parent node in T for
x ∈ {a, b, c}. Then a′ , b′ , c′ are pairwise different, and since ab ∈ E, bc ∈ E and ac ∈ E,
i.e., their distance in T is at most 3, a′ b′ , b′ c′ and c′ a′ must be edges in T , i.e., the tree
T contains a C3 which is a contradiction. This shows Claim 11.4.1. ⋄
Obviously, the sets Vi , i ∈ {1, . . . , k}, are clique modules in G. If G does not fulfill the
assumption that for all i ∈ {1, . . . , k}, |Vi | = 1, then let G∗ be the induced subgraph of
G by taking a representative vertex from each of the nonempty sets Vi , i ∈ {1, . . . , k}.
For G∗ , Claim 11.4.1 applies, i.e., G∗ is a tree. Now it is easy to see that G is the result
of substituting the cliques Vi into the corresponding representative vertices in G∗ .
Conversely, assume that G is the result of substituting the vertices of a suitable tree G∗
by some cliques. Then by Observation 11.2, G∗ is a 3-leaf power, and by Proposition
11.5, G is also a 3-leaf power which shows Theorem 11.10.
2
By Theorem 11.10, it is obvious that 3-leaf powers are distance-hereditary. Recall that
by Theorem 8.10, a graph is distance-hereditary and chordal if and only if it is gem-free
chordal, and by Theorem 8.6, a connected graph G is distance-hereditary if and only
if G can be generated from a single vertex by repeatedly adding a pendant vertex, a
false twin or a true twin. For 3-leaf powers, one can show the following:
Corollary 11.6 A connected graph G is a 3-leaf power if and only if G is the result
of a sequence of pendant vertex operations, starting with a single vertex, followed by a
sequence of true twin operations.
132
bull
dart
gem
Figure 20: bull, dart and gem have no 3-leaf root
Proof. First assume that G is a 3-leaf power. Then, by Theorem 11.10, G is the
result of substituting cliques into the vertices of a tree T . Let σ1 be a sequence of
pendant vertex operations which generate T . Now the substitution of cliques into T
which produces G can be done by a sequence σ2 of true twin operations. Thus, G
results from a sequence of pendant vertex operations followed by a sequence of true
twin operations.
Conversely assume that G results from a sequence σ1 of pendant vertex operations,
followed by a sequence σ2 of true twin operations. Then σ1 generates a tree T , and
σ2 produces a set of cliques which are substituted into the vertices of T . Again by
Theorem 11.10, G is a 3-leaf power.
2
Note that if pendant vertex and true twin operations are mixed then also graphs can
be generated which are no 3-leaf powers; the bull and the dart are such examples. In
particular, the class of 3-leaf powers is not closed under the pendant vertex operation.
Corollary 11.6 can also be formulated in the following way:
Corollary 11.7 A graph G is a 3-leaf power if and only if every induced subgraph of
G is a forest or has true twins.
The following Theorem 11.11 which characterizes 3-leaf powers in terms of forbidden
subgraphs was already given in [184] using the notion of critical cliques. The proof
below given in [85] is simpler. For bull, dart and gem see Figure 20.
Theorem 11.11 A graph G is a 3-leaf power if and only if G is (bull, dart, gem)-free
chordal.
Proof. If G is a 3-leaf power then by Theorem 11.10, its connected components result
from a tree by substituting cliques into its vertices. It is easy to verify that bull, dart
and gem have no 3-leaf root, and since induced subgraphs of 3-leaf powers are 3-leaf
powers, G is (bull,dart,gem)-free chordal.
Conversely assume that G is (bull,dart,gem)-free chordal. We use Corollary 11.7 in
order to show that G is a 3-leaf power. It suffices to prove that G itself is a forest
or has true twins since all induced subgraphs of G are (bull,dart,gem)-free chordal.
Suppose that G is not a forest. Then, as G is chordal, G has a maximal clique Q with
133
at least three vertices. If Q has no neighbor in G − Q then clearly, every two vertices
in Q form true twins. Now, let v ∈ G − Q be adjacent to a vertex q1 ∈ Q. As Q is a
maximal clique, there is a vertex q2 ∈ Q such that v is nonadjacent to q2 . Consider a
vertex q3 ∈ Q \ {q1 , q2 }.
Now, if v is nonadjacent to q3 then q2 , q3 form true twins since otherwise, if a vertex u
distinguishes q2 and q3 , say u is adjacent to q2 and nonadjacent to q3 then u, v, q1 , q2 , q3
would induce a bull or dart or gem, or u, v, q1, q2 would induce a C4 (depending on the
possible edges uq1 , uv).
If v is adjacent to q3 then q1 , q3 form true twins since otherwise, if a vertex u distinguishes q1 and q3 , say u is adjacent to q1 and nonadjacent to q3 then u, v, q1, q2 , q3 would
induce a dart or gem, or u, v, q2 , q3 would induce a C4 (depending on the possible edges
uq2 , uv).
2
Theorem 11.12 below collects various characterizations of 3-leaf powers from [85, 184,
391].
Theorem 11.12 For a connected graph G, the following conditions are equivalent:
(i) G is a 3-leaf power.
(ii) G is (bull, dart, gem)-free chordal.
(iii) G results from substituting cliques into the vertices of a tree.
(iv) The critical clique graph CC(G) of G is a tree.
(v) G results from adding pendant vertices, starting with a single vertex, followed by
adding true twins.
(vi) Every induced subgraph of G is a forest or has true twins.
See [85, 391] for more structural details and other equivalent conditions, and in particular [85] for a simple linear time recognition algorithm of 3-leaf powers.
11.4.3
New characterizations of squares of trees
Powers of graphs, and in particular, squares and other powers of trees is a well-studied
topic in graph theory [265, 405]. Efficient algorithms for recognizing squares of trees
were studied in [300, 317, 334]. Lin and Skiena [334], and Lau [317] give a linear time
algorithm for recognizing whether a given graph is the square of a tree. In [129], a
linear time algorithm for deciding whether a graph is the (k-th) power of a tree is
given.
In this section, we provide new structural characterizations for squares of trees which
we use later as tools for characterizing 4-leaf powers. As a direct consequence, we
obtain a new linear time algorithm for recognizing squares of trees and computing the
tree root.
134
Recall that Kp− denotes the clique with p vertices minus an edge. For tree T = (V, ET )
let Tx denote the star with center x in T , i.e., Tx = NT [x].
Observation 11.3 ([94]) Let G = T 2 for a tree T . Then the following conditions are
equivalent:
(i) {x, y} is a 2-cut in G.
(ii) xy is the edge shared by the two triangles in a diamond of G.
(iii) x and y are no leaves in T and x, y ∈ Tx ∩ Ty .
(iv) xy is the mid-edge of a P4 in T .
Proof. (i) ⇒ (ii): It is well known that G = T 2 for a tree T is 2-connected and
chordal. Consider two connected components A and B of G − {x, y}. As G is chordal
(and every 2-cut in G is a minimal cutset), there exist vertices a ∈ A and b ∈ B such
that a and b are adjacent to x and y, hence (ii).
(ii) ⇒ (iii): Let a, b, x, y induce a diamond in G with edges ax, ay, bx, by and xy. Since
leaves in T are clearly simplicial vertices in G, x and y are no leaves in T . So, (iii)
follows if xy is an edge of T . Suppose to the contrary that xy 6∈ E(T ). Then there
exists a vertex z such that zx and zy are edges of T . Now, since a and b are adjacent
to x and y in G, a and b must be adjacent to z in T , or one of them is adjacent to
z in T and the other is the vertex z. In any case, a and b would be adjacent in G, a
contradiction. Thus, xy ∈ E(T ), hence (iii).
(iii) ⇒ (iv): Obvious by definition of Tx , Ty .
(iv) ⇒ (i): Let X and Y be the vertex sets of the subtrees of T − xy containing x,
respectively, y. Since x and y are no leaves in T , X − x and Y − y are nonempty.
Clearly, any path in G connecting a vertex in X − x and a vertex in Y − y must contain
x or y, hence {x, y} is a 2-cut of G.
2
The following fact has also been shown in [317]; in [94] we give another short proof. A
family of subsets of a ground set has the Helly property if for every subfamily, pairwise
nonempty intersection implies nonempty total intersection. Recall that subtrees of a
tree fulfill the Helly property.
Observation 11.4 Let T be a tree. Then the maximal cliques in T 2 are exactly the
stars Tx , x ∈ V (T ), for which x is no leaf in T .
Proof. If x is a non-leaf vertex of T , Tx clearly induces a maximal clique in T 2 .
Conversely assume that Q = {q1 , . . . , qk } is a maximal clique in T 2 . We have to show
that there is a vertex x such that Q = Tx . The property that Q is a clique means that
Tqi ∩ Tqj 6= ∅ for all i, j ∈ {1, . . . , k}. Since subtrees of a tree have the Helly property,
this means that there is a vertex x in the intersection of all Tqi , i ∈ {1, . . . , k}, i.e.,
Q = Tx .
2
135
G1
G2
G3
G6
G5
G4
G7
G8
Figure 21: Graphs G1 − G8
For the graphs G1 , . . . , G5 mentioned in Theorem 11.13, see Figure 21.
In [94], we have characterized the squares of trees as follows:
Theorem 11.13 For every graph G, the following conditions are equivalent:
(i) G is the square of a tree.
(ii) G is chordal, 2-connected and has the following properties:
(1) Every pair of distinct maximal cliques has at most two vertices in common.
(2) Every 2-cut belongs to exactly two maximal cliques of G.
(3) Every pair of nondisjoint 2-cuts belongs to the same maximal clique.
(4) All 2-cuts contained in the same maximal clique of G have a common vertex.
(iii) G is chordal, 2-connected, and (G1 , . . . , G5 )-free.
Proof. (i) ⇒ (ii): Let G = T 2 for a tree T . Then by Observation 11.4, the maximal
cliques of G are exactly the stars in T whose midpoints are no leaves in T . Clearly, for
x 6= y, |Tx ∩ Ty | ≤ 2. This shows (1) of (ii).
If |Tx ∩ Ty | = 2 then, by Observation 11.3, Tx ∩ Ty = {x, y} is a 2-cut and vice versa
which shows (2). Moreover if {u, x} and {x, y} are 2-cuts having the common vertex x
then both are contained in the maximal clique Tx which shows (3). Finally all 2-cuts
in the same maximal clique, say Tx , have x in common which shows (4).
(ii) ⇒ (i): Let G satisfy (ii). If G is a clique then G is the square of a star and (i)
holds. Now assume that G is not a clique but chordal and 2-connected. Then let R be
a clique tree for G (see [93, 414] for more details on clique trees of chordal graphs and
[414] for a simple linear time algorithm for determining such a tree). We construct a
spanning tree T of G bottom-up along R as follows.
For each 2-cut {x, y} of G belonging to the two maximal cliques A and B, let x be the
common vertex of all 2-cuts in A (if A has more than one; cf. condition (4)). Note
136
that for maximal cliques which are adjacent in R, the intersection contains exactly two
vertices since G is 2-connected and satisfies condition (1). Then, by condition (3), y
is the common vertex of all 2-cuts in B (if B has more than one.) Add xy to E(T )
and add the edges xa, a ∈ A − x, and the edges yb, b ∈ B − y, to E(T ). Clearly, T
is a spanning tree of G. Moreover, by construction and by condition (2), T has the
following property:
(∗) For each 2-cut {x, y} belonging to maximal cliques Q and Q′ , T ∩ Q is a star at
a vertex z ∈ {x, y} and T ∩ Q′ is a star at the other vertex in {x, y} \ {z}.
We now claim that G = T 2 . First, since every edge of G belongs to a maximal clique,
(∗) implies that E(G) ⊆ E(T 2 ). We show now that E(T 2 ) ⊆ E(G). Let uv be an
edge of T 2 not belonging to T . Then there exists a vertex x such that xu and xv are
edges of T , hence both xu and xv are edges of G. If uv is also an edge of G, we are
done. If not, as G is 2-connected, there exists an induced path u1 u2 . . . up , p ≥ 3, in
G − x connecting u1 = u and up = v. As G is chordal, x is adjacent to all ui . As G
satisfies condition (1), {x, ui }, 2 ≤ i ≤ p − 1, are 2-cuts of G. (Indeed, if there exists
an induced path ui−1 v1 . . . vq vq+1 in G − {x, ui } connecting ui−1 and vq+1 = ui+1 then,
as G is chordal, x and ui are adjacent to all v1 , . . . , vq . But then the maximal cliques
containing x, ui−1 , ui , v1 and x, ui , v1 , v2 , respectively, have more than two vertices in
common, contradicting condition (1).) Now, the maximal cliques Q and Q′ containing
x, u1 , u2 and x, u2 , u3, respectively, and the 2-cut {x, u2 } contradict (∗). Thus, we have
shown E(T 2 ) ⊆ E(G), hence G = T 2 .
(i) ⇒ (iii): If G = T 2 for some tree T then G is chordal and 2-connected. Moreover,
every 2-connected induced subgraph of T 2 is also the square of some tree. (Namely,
if the 2-connected induced subgraph H of T 2 is not a clique, then T ′ = T [V (H)] is a
subtree of T with H = T ′2 .) The 2-connected graphs G1 , . . . , G5 are not the square of
any tree since their maximal cliques do not satisfy one of the conditions (1)-(4) of (ii).
Thus, (iii) follows.
(iii) ⇒ (ii): Let G be a 2-connected (G1 , . . . , G5 )-free chordal graph. To establish
condition (1) of (ii), note that the intersection of two maximal cliques contains at most
two vertices since G is G3 -free. If condition (2) does not hold, then G would contain
G2 as an induced subgraph.
We now prove that G satisfies condition (3). Assume to the contrary that the 2-cuts
{x, y} and {x, z} of G do not belong to the same maximal clique. In particular, y
and z are nonadjacent. Let A be a connected component of G − {x, y} not containing
z, and let B be a connected component of G − {x, z} not containing y. Note that
B ⊆ G − (A ∪ {x, y}). As G is 2-connected and chordal, there exist vertices a ∈ A,
b ∈ B such that a is adjacent to x and y, and b is adjacent to x and z. Moreover, there
exists an induced path v1 v2 . . . vp , p ≥ 3, in G − x connecting v1 = y and vp = z, and x
is adjacent to all vi , 1 ≤ i ≤ p. By the choice of A and B, all vi , 2 ≤ i ≤ p − 1, do not
belong to A ∪ B, and a is nonadjacent to all vi , 2 ≤ i ≤ p, and b is nonadjacent to all
vi , 1 ≤ i ≤ p − 1. Thus, x, a, v1 , v2 , v3 together with v4 (if p ≥ 4) or with b (if p = 3)
induce a G4 in G, a contradiction. Thus, G satisfies condition (3).
137
Finally, we show that G satisfies condition (4). First, consider two 2-cuts {x, y} and
{x′ , y ′} in the same maximal clique of G. Suppose {x, y} ∩ {x′ , y ′} = ∅. Let A be
a connected component of G − {x, y} not containing x′ , y ′, and let B be a connected
component of G − {x′ , y ′} not containing x, y. As G is 2-connected and chordal, there
exist vertices a ∈ A, b ∈ B such that a is adjacent to x, y, and b is adjacent to x′ , y ′. By
the choice of A and B, a is nonadjacent to x′ , y ′, and b is nonadjacent to x, y. Hence
a, b, x, y, x′ and y ′ induce a G5 in G. Thus, we have shown that every two 2-cuts in
the same maximal clique must have a common vertex. Next, we show that no three
2-cuts in G form a triangle. For, if {x, y, z} induces a triangle, where {x, y}, {x, z}
and {y, z} are 2-cuts of G, then consider a connected component A of G − {x, y} not
containing z, a connected component B of G−{x, z} not containing y, and a connected
component C of G − {y, z} not containing x. As before, there exist vertices a ∈ A,
b ∈ B and c ∈ C such that a is adjacent to x, y and nonadjacent to z, b is adjacent
to x, z and nonadjacent to y, c is adjacent to y, z and nonadjacent to x, and a, b, c
are pairwise nonadjacent. That is, a, b, c, x, y and z induce a G1 in G, a contradiction.
Thus, we have shown that no three 2-cuts of G form a triangle. It follows that G
satisfies condition (4).
2
As testing chordality [404, 421] and determining a clique tree [414] can be done in
linear time, Theorem 11.13 implies the following linear time algorithm to test whether
a given graph G is the square of some tree T : Construct a candidate for tree T , test
whether T is a tree, and test whether G = T 2 . The latter can be done in linear time
(cf. [317, 334]). Note that the linear time algorithm for recognizing squares of trees
given in [334] builds the tree root incrementally, by identifying the leaves and their
parents of a possible root tree, and repeating the process recursively, while in [317] this
is done by reducing the problem to recognizing the squares of trees with a specified
neighborhood. In contrast, our algorithm deduces the tree root directly from the clique
structure of the square of a tree.
Corollary 11.8 ([317, 405]) The tree roots of squares of trees are unique up to isomorphism.
11.4.4
Structure of (basic) 4-leaf powers
The following characterization of 4-leaf powers by Rautenbach [391] inspired the results
in the previous section on squares of trees, and in particular Theorem 11.13 (for G1 -G8
see Figure 21).
Theorem 11.14 ([391]) Let G be a graph without true twins. Then G is a 4-leafpower if and only if G is chordal and (G1 , . . . , G8 )-free.
Subsequently we study 4-leaf powers in more detail. By Proposition 11.5, every 4-leaf
power results from substituting cliques into the vertices of a basic 4-leaf power. Thus,
a new characterization of basic 4-leaf powers automatically leads to a new characterization of 4-leaf powers in general. In [94] we have characterized the 2-connected basic
4-leaf powers as follows:
138
Proposition 11.7 For every graph G, the following conditions are equivalent:
(i) G has a basic 4-leaf root without invisible vertices;
(ii) G is the square of some tree;
(iii) G is a 2-connected basic 4-leaf power.
Proof. (i) ⇔ (ii): follows by Proposition 11.6 (i).
(ii) ⇒ (iii): Let G = T 2 for some tree T . Then, G is 2-connected, and by Proposition 11.6 (i), G is a basic 4-leaf power.
(iii) ⇒ (ii): Let G be a 2-connected basic 4-leaf power, and R be a basic 4-leaf root
of G. If R has no invisible vertices, (ii) can be deduced using (i). In case R has an
invisible vertex, we claim that G must be a clique (and we are done). Suppose v is an
invisible vertex of R. As G is connected, no neighbor of v in R is an invisible vertex.
Root R at v, and for any neighbor w of v in R let Rw denote the subtree of R rooted
at w. Now, if G is not a clique, then there exists a neighbor w of v in R such that Rw
has a leaf x different from the leaf y adjacent to w (since w is not an invisible vertex
and R is a basic 4-leaf root, w has exactly one neighbor that is a leaf). Let w ′ 6= w be
another neighbor of v in R. Clearly, any path in G connecting x ∈ Rw and a vertex
that is a leaf in Rw′ must pass y, contradicting the 2-connectedness of G.
2
Theorem 11.13 together with Proposition 11.7 has some consequences.
Corollary 11.9 2-connected basic 4-leaf powers can be recognized in linear time, and
a basic 4-leaf root of a 2-connected basic 4-leaf power can be constructed in linear time.
Corollary 11.10 The graphs G1 , G2 , G3 , G4 , and G5 in Figure 21 are not basic 4-leaf
powers. Moreover, G1 and G4 are not 4-leaf powers at all.
Actually, G1 as 3-sun is even not strongly chordal.
Corollary 11.11 Every 2-connected basic 4-leaf power different from a clique has a
unique basic 4-leaf root, up to isomorphism.
In particular, the diamond has a unique basic 4-leaf root. This gives a simple proof for
the following fact:
Corollary 11.12 The graphs G6 , G7 , and G8 in Figure 21 are not basic 4-leaf powers.
Moreover, G6 is not a 4-leaf power at all.
Observation 11.5 Let p ≥ 3 be an integer and G be a (p−2)-connected chordal graph.
If G is not a clique, then every vertex of G is contained in some Kp− .
139
Proof. (Induction on p.) The case p = 3 is easily verified. Let p ≥ 4, and let G be a
(p − 2)-connected chordal graph. Suppose G is not a clique, and consider an arbitrary
vertex v of G. If v was a universal vertex, then G − v is a (p − 3)-connected chordal
−
graph that is not a clique. By induction, G − v contains a Kp−1
which together with
−
v forms a Kp . Therefore, we can assume that v has a non-neighbor in G. Let A be
a connected component of G − N[v], S ⊆ N(v) be the set that minimally separates v
from A, and w ∈ A be a vertex that has the most neighbors in S. As G is chordal, S
induces a clique (it is well known that every minimal cutset in a chordal graph induces
a clique). We claim that w is adjacent to every vertex in S. Suppose to the contrary
that x ∈ S is a vertex that is not adjacent to w. Let P be a shortest path from x to w
such that every vertex of P , except x, is in A. Clearly, P has at least 3 vertices. Let
y be the vertex next to x on P . By choice of w, there exists a vertex z ∈ S that is
adjacent to w but not to y. Then, the subgraph of G induced by the vertices of P and
z has a chordless cycle on at least 4 vertices, a contradiction. Now, the connectivity of
G implies that |S| ≥ p − 2. Therefore, v, w, and any p − 2 vertices of S induce a Kp− .
2
In the proof of Theorem 11.15 below we need the following tree structure of the blocks
in connected graphs. The block-cut-vertex graph BC(G) of a graph G has all blocks
and cut vertices of G as vertices, and two vertices in BC(G) are adjacent if one of them
is a block in G and the other is a cut vertex in G belonging to the block. It is well
known (cf. [265]) that the block-cut-vertex graph BC(G) of any connected graph G is
a tree, called the block-cut-vertex tree of G.
In [94], we have characterized the basic 4-leaf powers as follows:
Theorem 11.15 For every connected graph G, the following conditions are equivalent:
(i) G is a basic 4-leaf power.
(ii) Every block of G is the square of some tree, and for every non-disjoint pair of
blocks, at least one of them is a clique.
(iii) G is an induced subgraph of the square of some tree.
(iv) G is chordal and (G1 , . . . , G8 )-free.
Proof. (i) ⇒ (ii): Let G be a basic 4-leaf power. Then, by Proposition 11.7, every
block of G is the square of some tree. Suppose G had non-disjoint blocks A and B
such that neither A nor B is a clique. Let u be the cut vertex shared by A and B. By
Observation 11.5, u belongs to a diamond in each of A and B but then G contains G6 ,
G7 , or G8 as an induced subgraph, a contradiction to Corollary 11.12.
(ii) ⇒ (i): We will construct a basic 4-leaf power R for G from appropriately constructed
basic 4-leaf roots RB for blocks B of G. Let B be a block of G with vertices x1 , . . . , xk ,
k ≥ 2.
140
If B is a clique, take new vertices SB , X1 , . . . , Xk and form a star with center vertex
SB and edges SB Xi , 1 ≤ i ≤ k. Then RB is obtained from this star by attaching leaf
xi to node Xi , 1 ≤ i ≤ k.
If B is not a clique, then consider the tree TB (with new vertices) such that B is
isomorphic to TB2 . Let Xi be the node of TB corresponding to the vertex xi of B. Then
RB is obtained from TB by attaching leaf xi to node Xi (cf. proof of Proposition 11.6).
The basic 4-leaf root R for G is now obtained by putting all RB together as follows.
For each cut vertex v of G, let B1 , . . . , Bk be the neighbors of v in the block-cut-vertex
tree BC(G) of G. Identify the parents of v from the roots RBi and delete all but one
copy of leaf v. Let the resulting graph be R. As BC(G) is a tree, R is a tree.
We claim that R is a basic 4-leaf power for G. It is clear from the construction of R that
the adjacencies within a block of G are preserved by R. Consider vertices x ∈ B\B ′
and y ∈ B ′ \B, where B and B ′ are two distinct non-disjoint blocks of G. As at least
one of B, B ′ is a clique, from the construction of R, dR (x, y) ≥ 5.
(i) ⇔ (iii): Follows from Corollary 11.5.
(i) ⇒ (iv): By Corollaries 11.10 and 11.12, and Proposition 11.2.
(iv) ⇒ (ii): By Theorem 11.13, every block of G is the square of a tree. The rest of
the implication follows from Observation 11.5 and the fact that G is (G6 , G7 , G8 )-free.
2
Corollary 11.13 Basic 4-leaf powers can be recognized in linear time, and a basic
4-leaf root of a basic 4-leaf power can be constructed in linear time.
Recall that in Theorem 11.14, 4-leaf powers without true twins are characterized. Note
that each 4-leaf power without true twins is a basic 4-leaf power but not vice versa (cf.
the comments after Proposition 11.6). The equivalence (i) ⇔ (iv) in Theorem 11.15
extends the characterization of 4-leaf powers without true twins in Theorem 11.14 to
the larger class of basic 4-leaf powers. Theorem 11.15 can be interpreted as follows:
The graphs G1 , . . . , G5 are responsible for the structure of the 2-connected components whereas the graphs G6 , G7 , G8 represent the gluing conditions of the 2-connected
components, i.e., blocks. Note that all forbidden subgraphs G1 − G8 express certain
separator properties.
Based on the characterization of basic 4-leaf powers, in [94], we give a characterization
of 4-leaf powers in general and a linear time recognition algorithm for them.
11.4.5
Structure of (basic) distance-hereditary 5-leaf powers
The results in this section are from [92]. Recall that by Theorem 8.10, a chordal graph
is distance-hereditary if and only if it is gem-free.
(i) A plump dart is the graph resulting from the dart by replacing each of the vertices
of degree 1 or degree 2 by a nonempty union of cliques, the vertex of degree 3 by
a nonempty clique, and the vertex of degree 4 by a K2 .
141
(ii) A plump bull is the graph resulting from the bull by replacing each of the two
cutvertices by a K2 , the vertices of degree 1 by a nonempty union of cliques, and
the vertex of degree 2 by a nonempty clique.
F1
F2
F4
F6
F3
F5
F8
F7
Figure 22: Forbidden induced subgraphs F1 , . . . , F8 of 2-connected basic distancehereditary 5-leaf powers
Two-connected distance-hereditary basic 5-leaf powers can be characterized as follows
(for the graphs F1 , . . . , F8 see Figure 22).
Theorem 11.16 ([92]) Let G be a 2-connected distance-hereditary graph. Then the
following statements are equivalent:
(i) G is a basic 5-leaf power.
(ii) G is (F1 , . . . , F8 )-free chordal.
(iii) G is a plump dart or a plump bull or a 3-leaf power.
Similarly as for 4-leaf powers, also for 5-leaf powers, gluing conditions for blocks play
an important role. The forbidden subgraphs F9 , . . . , F34 (which we do not define here
- see [92] for them) reflect these conditions.
A vertex v is a special vertex in graph G if NG (v) is a clique module. A vertex is an
inner vertex in a plump dart if it results from replacing a degree 3 or degree 4 vertex
in the dart. It is an inner vertex in a plump bull if it results from replacing a degree 2
or degree 3 vertex in the bull.
Theorem 11.17 ([92]) For any distance-hereditary graph G, the following statements
are equivalent:
142
(i) G is a basic 5-leaf power.
(ii) G is (F1 , . . . , F34 )-free chordal.
(iii) (a) The blocks of G are 3-leaf powers, plump darts or plump bulls.
(b) For every two blocks B 6= B ′ of G with B ∩ B ′ = {v}, if B is not a 3-leaf
power and v is an inner vertex of B then B ′ is a 3-leaf power and v is a
special vertex of B ′ .
(c) Every block that is a clique of order at least three contains at most one
cutvertex that is an inner vertex of another block.
Corollary 11.14 ([92]) Distance-hereditary basic 5-leaf powers can be recognized in
linear time, and a basic 5-leaf root of a distance-hereditary basic 5-leaf power can be
constructed in linear time. Moreover, distance-hereditary 5-leaf powers can be recognized in linear time.
It remains an open problem to characterize 5-leaf powers in general. The characterization of distance-hereditary basic 5-leaf powers in Theorem 11.17 is a promising first
step. Note that in [128], a linear time recognition of 5-leaf powers is given.
11.5
Variants and generalizations of leaf powers
In this section, we present two variants as well as a generalization of leaf powers,
namely simplicial powers. The variants modify the distance conditions in root trees,
and simplicial powers generalize root trees to general host graphs and leafs to simplicial
vertices.
11.5.1
On (k, ℓ)-leaf powers
In [105], we defined the following natural variant of k-leaf powers and k-leaf roots: Let
G = (V, E) be a finite simple graph. Let 2 ≤ k < ℓ for integers k and ℓ. A tree T
is a (k, ℓ)-leaf root of G if V is the set of leaves of T , for all edges xy ∈ E, we have
dT (x, y) ≤ k, and, for all non-edges xy 6∈ E, dT (x, y) ≥ ℓ holds. G is a (k, ℓ)-leaf power
if it has a (k, ℓ)-leaf root.
Thus, every k-leaf power is a (k, k + 1)-leaf power, and every (k, ℓ)-leaf power is an
(i, j)-leaf power, for all pairs (i, j) with k ≤ i < j ≤ ℓ. In particular, every (k, ℓ)-leaf
power is a k ′ -leaf power, for all k ′ with k ≤ k ′ ≤ ℓ − 1.
For instance, every (4, 6)-leaf power is a k-leaf power for all k ≥ 4. Strictly chordal
graphs which are originally defined in [304] as those chordal graphs whose clique hypergraph is a so-called strict hypertree are exactly the (dart,gem)-free chordal graphs
according to Corollary 2.2.2. in [302]:
Proposition 11.8 A graph is strictly chordal if and only if it is (dart, gem)-free
chordal.
143
Proposition 11.9 For every graph G and for every k ≥ 2 and ℓ > k, G is a (k, ℓ)-leaf
power if and only if every graph resulting from G by substituting its vertices by cliques
is a (k, ℓ)-leaf power.
Proof. If T is a (k, ℓ)-leaf root for the (k, ℓ)-leaf power G = (V, E), and G′ is the
result of substituting a clique Q into a vertex u ∈ V , then attach all vertices in Q at
the same parent in T as u and skip u; the resulting tree T ′ is a (k, ℓ)-leaf root for G′ .
The converse direction obviously holds.
2
The subsequent Theorem 11.18 has been our motivation for defining and investigating
the notion of (k, ℓ)-leaf powers in [105].
Theorem 11.18 For a connected graph G = (V, E), the following conditions are equivalent:
(i) G is a (4, 6)-leaf power.
(ii) G is (dart, gem)-free chordal (i.e., strictly chordal).
(iii) G results from substituting cliques into the vertices of a block graph.
(iv) The critical clique graph CC(G) of G is a block graph.
(v) G is chordal, and the pairwise intersections of maximal cliques in G are pairwise
disjoint or equal.
(vi) G is chordal, and the pairwise intersections of maximal cliques in G are clique
modules in G.
Proof. (i) ⇒ (ii): Let G be a (4,6)-leaf power with (4,6)-leaf root T . Suppose that G
contains dart or gem as an induced subgraph. Then there are five vertices a, b, c, d, e
in V (G) such that {a, b, c, d} induce a diamond with bd ∈
/ E and e distinguishing a
and c, say ae ∈ E and ce ∈
/ E. According to Buneman’s condition (see Theorem 8.12)
applied to T , we have:
dT (a, c) + dT (b, d) ≤ max{dT (a, b) + dT (c, d), dT (a, d) + dT (b, c) ≤ 8}
since ab, ad, bc, cd ∈ E. As bd ∈
/ E, we have dT (b, d) ≥ 6. Thus dT (a, c) ≤ 2, i.e.,
the leaves a and c have the same parent node a′ in T . Furthermore, dT (a, e) ≤ 4 and
dT (c, e) ≥ 6. Then
6 ≤ dT (c, e) ≤ dT (c, a′ ) + dT (a′ , e) ≤ 1 + 3
which is a contradiction. Hence, G is (dart,gem)-free. Moreover, G, as a (4,6)-leaf
power, is chordal.
(ii) ⇒ (iv): Let G be (dart,gem)-free chordal. We claim that G′ = CC(G) is a block
graph. As an induced subgraph of G, G′ = (V ′ , E ′ ) is (dart,gem)-free chordal. By
144
Theorem 8.13 it suffices to show that G′ is diamond-free. Suppose to the contrary that
the vertices a, b, c, d induce a diamond in G′ with bd ∈
/ E ′ . As G′ has no nontrivial
clique modules (implying that {a, c} is not a module in G′ ), there is a vertex z ∈ V ′
distinguishing a and c, say az ∈ E ′ and cz ∈
/ E ′ . Since G′ is dart-free, z sees b or d,
and since G′ is gem-free, z sees both of them but now z, b, c, d induce a C4 in G′ which
is a contradiction. Thus, G′ is a block graph.
(iv) ⇒ (iii): By definition of its critical clique graph CC(G), G results from CC(G) by
substituting the corresponding clique modules into the vertices of CC(G) which in (iv)
is supposed to be a block graph.
(iii) ⇒ (i): Let G result from substituting cliques into the vertices of a block graph
G′ = (V ′ , E ′ ). By Proposition 11.9, it suffices to show that G′ is a (4, 6)-leaf power. We
construct a (4, 6)-leaf root T of G′ as follows: For every block B of G′ , we construct a
subtree TB by first taking a star whose center is a new node cB and whose leaf set is the
set of vertices of B. Then each pendant edge is subdivided by exactly one node. Now
if B and B ′ are two blocks of G such that B and B ′ have a cut vertex v in common,
then identify v and its parent vertices in TB and TB′ . This construction guarantees
that the T -distance of vertices x, y ∈ V with xy ∈ E is at most 4 and the T -distance
of vertices x, y ∈ V with xy ∈
/ E is at least 6, so that T is a (4, 6)-leaf root of G′ .
(iii) ⇒ (vi): Let G result from substituting cliques into the vertices of a block graph
B. Clearly, G is chordal. Substituting a clique Q into a vertex v of B which is not a
cut vertex of B only enlarges the block containing v while substituting Q into a cut
vertex v creates a clique module which is the pairwise intersection of blocks containing
cut vertex v. Now it is easy to see that every pairwise intersection of maximal cliques
in G is a clique module in G.
(vi) ⇒ (v): Suppose that all pairwise intersections of maximal cliques in G are clique
modules but there are intersections Q ∩ Q′ , R ∩ R′ of maximal cliques in G which are
neither equal nor disjoint. Let without loss of generality, x ∈ (Q ∩ Q′ ) \ (R ∩ R′ ) and
y ∈ (R ∩R′ ) ∩(Q∩Q′ ). Then, without loss of generality, x ∈
/ R and y ∈ R which means
that there is a third vertex z ∈ R with yz ∈ E and xz ∈
/ E, thereby distinguishing x
′
and y contradicting that Q ∩ Q is a module.
(v) ⇒ (iii): Suppose that G is connected and chordal such that the pairwise intersections of maximal cliques in G are pairwise disjoint or equal. We claim that the
pairwise intersections of maximal cliques in G are modules. Suppose to the contrary
that, for the maximal cliques Q and Q′ , Q ∩ Q′ is not a module, i.e., there are vertices
x, y ∈ Q ∩ Q′ and a vertex z ∈
/ Q ∩ Q′ distinguishing x and y, say zx ∈ E and zy ∈
/ E.
′
′′
Then z ∈
/ Q ∪ Q , and there is a maximal clique Q containing z and x. Since y ∈
/ Q′′ ,
′
′′
′
′′
Q ∩ Q 6= Q ∩ Q but Q ∩ Q intersects Q ∩ Q , a contradiction which shows the claim.
Now contract each nonempty intersection of maximal cliques to one vertex and denote
the resulting graph by G′ . Being isomorphic to a subgraph of G, G′ is chordal. We
claim that G′ is diamond-free. Suppose to the contrary that G′ contains a diamond with
vertices a, b, c, d such that ad ∈
/ E ′ . Then in G there are maximal cliques Q containing
a, b, c and Q′ containing b, c, d such that b, c ∈ Q ∩ Q′ but in G′ , the intersection was
contracted to one vertex, a contradiction. It follows from Theorem 8.13 that G′ is a
145
block graph, and substituting the nonempty intersections of maximal cliques in G into
the vertices of G′ gives G.
2
The equivalence of conditions (ii) and (iii) in Theorem 11.18 was shown already in [86].
The equivalence of (ii) and (iv) is implicitly mentioned in [304] – Lemma 2.4 of [304]
says that G is strictly chordal if and only if in the critical clique graph CC(G) of G,
the nodes of every simple cycle form a clique. Note that the diamond is a simple cycle
which is not a clique and thus CC(G) is diamond-free chordal, i.e., CC(G) is a block
graph.
The same class, namely (dart,gem)-free chordal graphs, has been characterized in terms
of contour vertices and convexity properties in [116].
Corollary 11.15 Strictly chordal graphs are 4-leaf powers and 5-leaf powers ( and thus
also k-leaf powers for all k ≥ 4).
Corollary 11.15 is one of the main results (namely Theorem 4.1) in [304]. It has also
been mentioned in Theorem 2.5 of [304] that strictly chordal graphs can be recognized
in linear time. The proof in [304] is based on a linear time algorithm for constructing
the critical clique graph CC(G) for a given chordal graph G (see section 11.1).
By Theorem 11.18, the linear time recognition of strictly chordal graphs given in [304]
can be simplified in the following way:
(1) construct CC(G);
(2) check whether CC(G) is a block graph; according to Theorem 11.18 (iv), this
recognizes strictly chordal graphs.
In [105], we give another, conceptually very simple, linear time algorithm for recognizing (4, 6)-leaf powers without constructing CC(G).
Corollary 11.16 (4, 6)-leaf powers (and thus also strictly chordal graphs) can be recognized in linear time.
Inspired by Theorem 11.14 and Theorem 11.15, we also found the following characterization of the class of (basic) (6,8)-leaf powers which is a natural superclass of 4-leaf
powers (see Figure 21 for G1 , G4 , G6 and Figure 23 for H1 − H6 ):
Theorem 11.19 The following conditions are equivalent:
(i) G is a basic (6, 8)-leaf power.
(ii) G is (G1 , G4 , G6 , H1 , H2 , H3 , H4 , H5 , H6 )-free chordal.
(iii) G is an induced subgraph of the square of some block graph.
146
H1
H2
H3
H5
H4
H6
Figure 23: Forbidden subgraphs H1 , . . . , H6 .
In particular, we have:
3-leaf powers ⊂ (4, 6)-leaf powers ⊂ 4-leaf powers ⊂ (6, 8)-leaf powers
In [105] we give many other properties of (k, ℓ)-leaf power classes and characterize
various other classes of this type.
11.5.2
Exact leaf powers
The results of this section are from [91]. G = (VG , EG ) is an exact k-leaf power if
there is a tree T = (VT , ET ) with set VG of leaves such that xy ∈ EG if and only if
dT (x, y) = k. Such a tree T is called an exact k-leaf root.
Requiring equality in the distance condition (instead of an upper bound k, as for k-leaf
powers) leads to huge differences in structural properties. Note that exact leaf powers
are not a special case of leaf powers.
house
A
domino
Figure 24: Forbidden subgraphs of exact 4-leaf powers
See Figure 24 for A and domino.
Theorem 11.20 ([91]) For a connected graph G, the following statements are
equivalent:
(i) G is an exact 3-leaf power.
147
(ii) G is (A, domino)-free chordal bipartite.
(iii) G is obtained from a tree by substituting vertices by stable sets.
(iv) Every induced subgraph of G is a forest or contains false twins.
(v) G is the result of a sequence of pendant vertex operations, starting with a single
vertex, followed by a sequence of false twin operations.
Corollary 11.17 Exact 3-leaf powers can be recognized in linear time, and an exact
3-leaf root of an exact 3-leaf power can be constructed in linear time.
The forbidden subgraphs of exact 4-leaf powers are given in Figure 24.
Theorem 11.21 ([91]) For a connected graph G, the following statements are
equivalent:
(i) G is an exact 4-leaf power.
(ii) G is hole-free and does not contain any graph from Figure 24 as an induced
subgraph.
(iii) G is obtained from a block graph by substituting vertices by stable sets.
Corollary 11.18 Exact 4-leaf powers can be recognized in linear time, and an exact
4-leaf root of an exact 4-leaf power can be constructed in linear time.
Characterizing exact k-leaf powers for k ≥ 5 is an open problem.
11.5.3
Simplicial powers of graphs
In [86] we define the following natural generalization of leaf powers as the key notion
of this section:
For integer k ≥ 1, graph G = (VG , EG ) is the k-simplicial power of graph H = (VH , EH )
if VG is the set of all simplicial vertices in H (in particular, VG ⊆ VH ) and for all
x, y ∈ VG with x 6= y,
xy ∈ EG ⇐⇒ dH (x, y) ≤ k.
Then such a graph H is a k-simplicial root of G.
If G is the k-simplicial power of H and if, in addition, VG consists of exactly the
degree-1 vertices, i.e., leaves of H, then we also say that G is the k-leaf power of H.
Note that in a bipartite graph, a simplicial vertex has degree 1, i.e., is a leaf. Thus, in
the sense of [380], leaf powers are exactly the simplicial powers of trees.
As the subsequent Proposition 11.10 shows, every graph is the simplicial power of some
graph. It is easy to see that a graph is the 1-simplicial power of some graph if and
only if it is the disjoint union of cliques, i.e., it does not contain an induced path P3
on three vertices. A graph is nontrivial if it has at least two vertices.
148
Proposition 11.10 ([86]) Every nontrivial graph is
(i) the 2-simplicial power of a split graph, and
(ii) the 4-leaf power of a bipartite graph.
Corollary 11.19 For all even k ≥ 4, every nontrivial graph is
(i) the k-simplicial power of a chordal graph and is
(ii) the k-leaf power of a bipartite graph.
The proof of Proposition 11.10 (i) constructs a split graph root which in general may
have exponentially many nodes. Thus it is natural to ask for a split graph root H with
minimum vertex number such that G is the 2-simplicial power of H. Let us consider
the following decision version of the problem.
2-simplicial split graph root
Instance: A graph G = (VG , EG ) and an integer k.
Question: Is there a split graph H = (VH , EH ) with |VH | ≤ k such
that G is the 2-simplicial power of H?
Theorem 11.22 2-simplicial split graph root is NP-complete.
The following Theorem 11.23 was the main motivation for introducing the concept of
simplicial powers.
Obviously, we have:
Observation 11.6 Let T be a tree.
(i) An edge xy of T is a simplicial vertex in the line graph L(T ) if and only if x or
y is a leaf in T .
(ii) Let ex = xx′ , ey = yy ′ be two edges of T where x′ and y ′ are leaves. Then
dL(T ) (ex , ey ) = dT (x′ , y ′ ) − 1.
Based on Theorem 8.16, we obtain:
Theorem 11.23 For k ≥ 2, a graph is a k-leaf power if and only if it is the (k − 1)simplicial power of a claw-free block graph.
Proof. ′′ =⇒′′ : Suppose that G = (VG , EG ) is a k-leaf power of a tree T = (VT , ET ).
To avoid triviality, we assume that T has more than one edge. By Theorem 8.16 and
Observation 11.6 (i), the line graph L(T ) is a claw-free block graph whose simplicial
vertices are in one-to-one correspondence to the end edges of T (as T has more than
149
one edge). For leaf x in T , let ex denote the T -edge containing x. Then by Observation
11.6 (ii),
xy ∈ EG ⇐⇒ dT (x, y) ≤ k ⇐⇒ dL(T ) (ex , ey ) ≤ k − 1
and ex , ey are simplicial in L(T ).
′′
⇐=′′ : Suppose that G = (VG , EG ) is the (k − 1)-simplicial power of a claw-free block
graph H. Then by Theorem 8.16, H = L(T ) for a tree T , and we may assume that T
has more than one edge.
Then each leaf v of T uniquely corresponds to the edge ev of T incident to v. By
Observation 11.6 (i), these edges ev of T are exactly the simplicial vertices of L(T ).
Moreover, for all distinct leaves v, v ′ we have dT (v, v ′) = dL(T ) (ev , ev′ ) + 1. Hence G is
the k-leaf power of tree T .
2
Corollary 11.20 For every t ≤ k + 1, each t-leaf power is the k-simplicial power of
some block graph.
Simplicial powers of (not necessarily claw-free) block graphs form an important subclass
of strongly chordal graphs of independent interest. It turns out that a graph is a ksimplicial power of a block graph if and only if it results from an induced subgraph of
the (k − 1)-th power of a block graph by substituting cliques into its vertices [86].
Recall that by Theorem 11.12, G is a 3-leaf power (of a tree) if and only if G is
(bull,dart,gem)-free chordal. Theorem 11.24 below characterizes the more general class
of 2-simplicial powers of block graphs as the (dart,gem)-free chordal graphs. Recall that
this class also appears as strictly chordal graphs and as (4,6)-leaf powers in Theorem
11.18.
Theorem 11.24 ([86]) For every graph G, the following statements are equivalent:
(i) G is the 2-simplicial power of a block graph.
(ii) G is (dart, gem)-free chordal.
(iii) G results by substituting cliques into the vertices of a block graph B.
Theorem 11.25 ([86]) G is the 2-simplicial power of a ptolemaic graph if and only
if it is ptolemaic.
Theorem 11.26 ([86]) For every graph G, the following statements are equivalent:
(i) G is the 2-simplicial power of a strongly chordal graph.
(ii) G is the 2-simplicial power of a strongly chordal split graph.
(iii) G is strongly chordal.
150
A graph is a basic k-simplicial power of a block graph if it admits a k-simplicial block
graph root in which each block contains at most one simplicial vertex. A maximal
clique Q in G = (V, E) is special if for all x, y ∈ V \ Q having a common neighbor in
Q, either N(x) ∩ Q = N(y) ∩ Q or |N(x) ∩ Q| = 1 or |N(y) ∩ Q| = 1. A vertex v ∈ V
is special if N[v] is a special clique in G.
Basic 3-simplicial powers of block graphs can be characterized in the following way:
Theorem 11.27 ([86]) The following conditions are equivalent for all graphs G.
(i) G is a basic 3-simplicial power of a block graph.
(ii) G is an induced subgraph of the square of a block graph.
(iii) Each block of G is a basic 3-simplicial power of a block graph, and each cut-vertex
v of G is non-special in at most one block containing v.
Recall that the same graph class is characterized in Theorem 11.19 in the context of
(6, 8)-leaf powers in a different way. In [86], we give many other results on simplicial
powers of graphs.
It is an open question whether every strongly chordal graph is a k-simplicial power of
a block graph for some k.
11.6
Pairwise compatibility graphs and leaf powers
G = (V, E) is a pairwise compatibility graph (PCG for short) [301] if there is a tree T
such that V is the set of leaves of T , there is a positive weight function w on the edges
of T and two non-negative real values dmin and dmax , dmin ≤ dmax , such that for any
two vertices x, y ∈ V (which are leaves in T ), xy ∈ E if and only if for the distance
dT,w (x, y) of x and y in T , dmin ≤ dT,w (x, y) ≤ dmax holds.
Obviously, PCG’s generalize leaf powers (by setting dmin = 0). The complexity of
recognizing PCG’s is an open problem; in [199], it is conjectured that this problem
is NP-complete. There are examples of graphs which are not PCG’s (initially, it was
believed that every graph is a PCG). See [117] for a survey on PCG’s.
The characterization of k-leaf powers for every k ≥ 5 remains a challenging open
problem. For 5-leaf powers, a linear time recognition algorithm was given in [128]. A
nice characterization of this class, however, is missing. For k ≥ 6, no characterization
and no efficient recognition of k-leaf powers is known. A characterization and efficient
recognition of leaf powers in general is also an open problem. In [375], based on
the structural results of [374] for strongly chordal graphs, another attempt is made
to solve this problem; among others, it results in more examples of strongly chordal
graphs which are not a leaf power.
Moreover, the complexity of determining the leaf rank of a given leaf power is an open
problem.
151
In [428], it is shown that the isomorphism problem for strongly chordal graphs is
isomorphism-complete, i.e., as hard as for general graphs. It might be interesting to
determine the complexity of the isomorphism problem for leaf powers.
Last but not least, it should be remarked that, starting with the result of Motwani and
Sudan [368] that it is NP-complete to recognize whether a given graph has a square
root (i.e., whether for given graph G there is a graph H such that G = H 2 ), recently,
there is tremendous work on the complexity of root problems of graphs and powers
of graphs – see e.g. [209, 317, 318, 320, 376]. Perhaps, techniques developed in these
papers might be useful for solving the open problems on leaf powers.
152
12
Complexity of Some Problems
Structured Graph Classes
on
Tree-
The most prominent classes with tree structure in this chapter are chordal and dually
chordal graphs, strongly chordal graphs and chordal bipartite graphs as well as distancehereditary graphs. In the following, we describe a variety of complexity results for some
problems on these and related classes. See also [414] for a final chapter on such results.
Distance-hereditary graphs are an exceptional case: Recall that for distance-hereditary
graphs, there is a long list of papers showing that certain problems are efficiently
solvable on this class. Most of these papers were published before the clique-width
aspect was found. Theorem 10.6 covers many of these problems; on the other hand,
it might be preferable to have direct dynamic programming algorithms using the tree
structure of distance-hereditary graphs since the constant factors in algorithms using
Theorem 10.6 are astronomically large (and similarly for graphs of bounded treewidth).
However, various problems such as Hamilton Cycle and variants cannot be expressed in
MSOL; see also the algorithm for Steiner tree on distance-hereditary graphs in section
8.4. The four basic problems can be solved in time O(n) if a pruning sequence of the
input graph is given [262].
12.1
Recognition
Recall that the recognition problem for chordal and dually chordal graphs is solvable in
linear time, while the recognition of strongly chordal and of chordal bipartite graphs can
be done in time O(min(n2 , m log n)) (see [414]). Recall also that distance-hereditary
graphs can be recognized in linear time [175, 262].
12.2
Independent Set and Chromatic Number
The four basic problems Independent Set [GT20], Clique [GT19], Chromatic Number
[GT4], and Partition into Cliques [GT15] (see [224]), are known to be polynomial-time
solvable for perfect graphs [251, 252] and thus for chordal graphs as well as strongly
chordal graphs. For chordal bipartite graphs, as a subclass of bipartite graphs, it is
simple. In some cases, there are better time bounds using prefect elimination orderings
and similar tools; recall the examples described in section 3.1.2 such as Frank’s linear
time algorithm for weighted Independent Set on chordal graphs.
For dually chordal graphs, however, these four problems are NP-complete [60]. In
[108, 310], MWIS is solved in polynomial time for AT-free graphs.
153
12.3
Hamiltonian Circuit
Hamiltonian Circuit ([GT37] of [224]) (HC for short) is NP-complete for strongly
chordal split graphs and for chordal bipartite graphs [370] (and thus it is NP-complete
for chordal as well as for dually chordal graphs).
For distance-hereditary graphs, HC was shown to be solvable in time O(n3 ) [377, 378],
in time O(n2 ) [287] and finally in time O(n + m); see [284, 286] for HC and variants
giving a unified approach. In [286], a detailed history of the complexity results for HC
on distance-hereditary graphs is given. For the subclass of bipartite distance-hereditary
graphs, a linear-time algorithm for HC was given already in [372].
The complexity of HC for AT-free graphs seems to be an open problem.
12.4
Dominating Set and Steiner Tree
Dominating Set [GT2] and Steiner Tree [ND12] [224] are solvable in linear time for
dually chordal graphs [60] and thus for strongly chordal graphs while they are NPcomplete for chordal graphs (even for split graphs [44]) and for chordal bipartite graphs
[371].
The Dominating set problem was solved in linear time in [134, 379] for distancehereditary graphs. The Efficient Domination and Efficient Edge Domination problems
are expressible in MSOL and thus efficiently solvable for distance-hereditary graphs.
12.5
Maximum Induced Matchings
For a graph G = (V, E), a subset M ⊆ E is an induced matching of G if the pairwise
G-distance of edges in M is at least 2, i.e., for e, e′ ∈ M with e 6= e′ , e ∩ e′ = ∅ and
there is no edge between e and e′ . In other words, the subgraph G[V (M)] is the disjoint
union of edges.
For a given graph G = (V, E), the Maximum Induced Matching problem (MIM for
short) asks for a maximum set of edges having pairwise distance at least 2.
Obviously, we have:
Proposition 12.1 M is an induced matching in G if and only if M is an independent
set in L(G)2 .
In [220, 241], induced matchings appear under the name strong matching. In [416], the
more general case of δ-separated matchings was considered; for δ = 2, a δ-separated
matching is the same as an induced matching.
While it is well known that the maximum matching problem is solvable in polynomial
time, the MIM problem is NP-complete even for bipartite graphs [119, 416]; in [416],
it is shown that for every δ ≥ 2, Maximum δ-Separated Matching is NP-complete for
154
bipartite graphs of maximum degree 4. Moreover, MIM is NP-complete for dually
chordal graphs [95] and for line graphs [309].
Chordal graphs are one of the first examples for a graph class where the MIM problem
is solvable in polynomial time via L(G)2 :
Theorem 12.1 ([119]) If G is a chordal graph then so is L(G)2 .
Proof. Let G = (VG , EG ) be a chordal graph. By Theorem 3.2, a graph is chordal if
and only if it is the intersection graph of a collection of subtrees of a tree. For G, let
F be such a collection of subtrees of tree T , i.e.,
uv ∈ EG ⇐⇒ Tu ∩ Tv 6= ∅
for Tu , Tv ∈ F . Note that EG is the vertex set of L(G)2 .
Claim. F ′ = {Tu ∪ Tv : uv ∈ EG } is a representation of L(G)2 as the intersection
graph of a family of subtrees of tree T .
Proof. First note that for every uv ∈ EG , Tu ∪ Tv is a tree since Tu ∩ Tv 6= ∅. Tu ∩ Tv
and Tw ∩ Tx are joined by an edge in the intersection graph of F ′ if and only if either
u or v is the same as w or x in which case the edges uv and wx are incident in G or
u, v, w, x are all distinct and (say) Tv ∩ Tw 6= ∅ in which case uv, vw, wx forms a path
of three edges in G if and only if uv and wx are joined by an edge in L(G)2 .
This shows the claim and thus, by Theorem 3.2, L(G)2 is chordal.
2
Theorem 12.2 ([241]) If G is a circular-arc graph then so is L(G)2 .
Theorem 12.3 ([242]) If G is a co-comparability graph then so is L(G)2 .
For various other similar results for intersection graphs such as interval graphs, polygoncircle graphs and, as a most general result, interval-filament graphs see [120] (interval
graphs and polygon-circle graphs are subclasses of interval-filament graphs):
Theorem 12.4 ([120]) If G is an interval-filament graph then so is L(G)2 .
Gavril [230] has shown that MWIS is solvable in polynomial time for interval-filament
graphs.
Corollary 12.1 MIM is solvable in polynomial time for interval-filament graphs.
Theorem 12.5 ([120, 126]) If G is an AT-free graph then so is L(G)2 .
AT-free graphs include co-comparability graphs, permutation graphs and trapezoid
graphs (see [93]). Recall that in [108, 310], MWIS is solved in polynomial time for
AT-free graphs.
155
Corollary 12.2 MIM is solvable in polynomial time for AT-free graphs.
Corollaries 12.1 and 12.2 generalize corresponding results for permutation graphs and
trapezoid graphs [242].
In [126], Theorem 12.5 is shown for the more general case of any power L(G)δ for δseparated matchings. Moreover, a linear time algorithm for finding a maximum induced
matching in a bipartite permutation graph (i.e., a bipartite AT-free graph) is given.
Theorem 12.6 ([242]) Finding a maximum induced matching on an interval graph
can be done in linear time.
In [442], it was shown that finding a maximum induced matching on trees can be done
in linear time. Both cases can be generalized to chordal graphs (avoiding the explicit
construction of L(G)2 for given input graph G):
Theorem 12.7 ([76]) Finding a maximum induced matching on an chordal graph can
be done in linear time.
12.5.1
MIM for hypergraphs
According to Proposition 12.1, we define induced matchings in a hypergraph H as
independent sets in L(H)2 .
For the MIM problem we have:
Theorem 12.8 ([95]) The MIM problem is solvable in polynomial time for α-acyclic
hypergraphs.
Proof. Let H = (V, E) be α-acyclic. Then by Corollary 5.3 (ii), L(H) is dually chordal.
By Corollary 5.3 (i), for a dually chordal graph G, its square G2 is chordal. Then by
Theorem 12.1, L(H)2 is chordal. Thus, the MIM problem for α-acyclic hypergraphs
can be solved by solving the MWIS problem for chordal graphs.
2
Theorem 12.9 ([95]) The MIM problem for hypertrees is NP-complete.
Proof. Let H = (V, E) be a hypertree. By definition, induced matchings in H correspond to independent sets in L(H)2 . By Theorem 4.4, L(H) is chordal. We claim:
For every chordal graph G, there is a hypertree H such that G ∼ L(H).
(10)
Proof. For a chordal graph G, the hypergraph C(G) of its maximal cliques is αacyclic and 2SEC(C(G)) ∼ G. Then by Proposition 2.2, C(G)∗ is a hypertree and
2SEC(C(G)) ∼ L(C(G)∗ ). Thus, H = C(G)∗ fulfills claim (10). ⋄
156
Thus, the MIM problem on hypertrees corresponds to the MIS problem on squares of
chordal graphs. The last problem, however, is NP-complete as the following reduction
shows:
Let G = (V, E) be any graph and let F = (V ∪ E ∪ {f, g}, EF ) be the graph with EF
consisting of all the edges of a clique E ∪ {f } and additionally a single edge f g, and all
edges between any edge e = xy ∈ E and its vertices x, y respectively. Thus, F is a split
graph and therefore, F is chordal. In G′ = F 2 , the vertex set V induces a subgraph
isomorphic to G. Let α(G) denote the maximum size of an independent set in G. We
claim:
α(G′ ) = α(G) + 1.
(11)
Proof. First assume that U is an independent set in G with |U| = α(G). Then
obviously, U ′ = U ∪ {g} is an independent set in G′ with |U ′ | = α(G) + 1. Moreover,
α(G′ ) ≤ α(G) + 1 since E ∪ {f, g} is a clique in G′ .
Conversely assume that U ′ is an independent set in G′ with |U ′ | = α(G) + 1. Since
E ∪ {f, g} is a clique in G′ , U ′ can contain at most one node of E ∪ {f, g}, i.e.,
|U ′ ∩ V | = α(G), and obviously, U ′ ∩ V is an independent set in G which shows (11). ⋄
This reduction shows Theorem 12.9.
2
12.6
Exact Cover, Efficient Domination, and Efficient Edge
Domination
12.6.1
Exact Cover
For a given hypergraph H = (V, E), the Exact Cover problem ([SP2] of [224]) asks for
the existence of a subset E ′ ⊆ E such that every vertex of V is in exactly one of the
sets in E ′ . The Exact Cover problem is NP-complete even for 3-regular hypergraphs
[298]. In [95], it is shown that the Exact Cover problem is NP-complete for α-acyclic
hypergraphs but solvable in linear time for hypertrees.
12.6.2
Efficient Domination
For a given graph G = (V, E), the Efficient Domination problem asks for the existence
of a set of closed neighborhoods of G forming an exact cover of V ; thus, the Efficient
Domination problem for G corresponds to the Exact Cover problem for the closed
neighborhood hypergraph of G. It was introduced by Biggs [46] under the name Perfect
Code.
In [24, 25], it was shown that the ED problem is NP-complete. It is known that ED
is NP-complete even for bipartite graphs [440], chordal graphs [440], line graphs of
bipartite graphs [347], planar bipartite graphs [348], chordal bipartite graphs [348],
and chordal unipolar graphs [412, 440].
As an example, we describe the reduction from Exact Cover to ED on chordal unipolar
157
graphs. A graph is unipolar if it can be partitioned into a clique and a P3 -free subgraph
(i.e., the disjoint union of cliques).
Theorem 12.10 ([412, 440]) The ED problem is NP-complete for chordal unipolar
graphs.
Proof. Let H = (V, E) be a hypergraph with V = {v1 , . . . , vn } and E = {e1 , . . . , em }.
We define the graph GH = (V ′ , E) as follows: Let V ′ := V ∪{x1 , . . . , xm }∪{y1 , . . . , ym }
and E := {vi vj | i 6= j} ∪ {xi yi | 1 ≤ i ≤ m} ∪ {xi v | v ∈ ei }. Clearly, GH is chordal
unipolar. (Note that the cliques in the P3 -free part of GH have size 2.)
We claim that H has an exact cover E ′ if and only if GH has an e.d. D:
If ei ∈ E ′ then xi ∈ D and if ei ∈
/ E ′ then yi ∈ D.
Obviously, E ′ is an exact cover of H if and only if D is an e.d. of GH .
2
In [95], we use the following relationship between ED on G and MWIS on G2 :
Lemma 12.1 Let G = (V, E) be a graph and w(v) := |N[v]| a vertex weight function
for G. Then the following are equivalent for any subset D ⊆ V :
(i) D is an efficient dominating set in G.
(ii) D is a minimum weight dominating set in G with w(D) = |V |.
(iii) D is a maximum weight independent set in G2 with w(D) = |V |.
In [71], this is extended to the vertex-weighted version WED of the ED problem. Let
α denote the matrix multiplication exponent; by [436], α < 2.3727.
Theorem 12.11 ([71]) Let C be a graph class for which the MWIS problem is solvable
in time T (|H|) on squares H of graphs from C. Then the WED problem is solvable on
graphs G ∈ C in time O(min{nm + n, nα } + T (|G2 |)).
For a dually chordal graph G, its square G2 is chordal (see Corollary 5.1). Thus, by
Lemma 12.1, for dually chordal graphs, ED is solvable in polynomial time. In [95], a
linear time solution for ED on dually chordal graphs is given. Thus, ED is solvable in
linear time for strongly chordal graphs while it is NP-complete for chordal bipartite
graphs [348].
For P6 -free chordal graphs, ED is solvable in polynomial time [70] by showing that for
a P6 -free chordal graph G with e.d., G2 is chordal.
Meanwhile, it was shown that WED is solvable in polynomial time for P6 -free graphs
[103, 338] thus reaching a dichotomy for ED on F -free graphs.
It is known that the WED (and consequently the ED) problem is solvable in polynomial
time on
158
− trees [439],
− co-comparability graphs [125, 133],
− split graphs [130],
− interval graphs [131, 133] and on their superclasses AT-free graphs [71],
− dually chordal graphs [71], and
− circular-arc graphs [131],
− permutation graphs [326],
− trapezoid graphs [326, 329],
− bipartite permutation graphs [348],
− convex bipartite graphs [95] and on their superclass interval bigraphs [95],
− distance-hereditary graphs [348],
− block graphs [440] and, more generally,
− graphs of bounded clique-width [159], which also include the hereditary efficiently
dominatable graphs [361].
In [99], we generalize the result for split graphs to 2P2 -free graphs (recall that by
Theorem 3.11, a graph is a split graph iff it is (2P2 , C4 , C5 )-free): WED is solvable in
linear time for 2P2 -free graphs. In [57], this result is generalized to P5 -free graphs.
12.6.3
Efficient Edge Domination
For a given graph G = (V, E), the Efficient Edge Domination problem is the Efficient
Domination problem for the line graph L(G). It appears under the name Dominating
Induced Matching problem in various papers; see e.g. [123]. See [82, 102, 123, 249, 346,
347] for various NP-completeness results for EED.
Observation 12.1 ([82, 102]) Let M be an e.e.d. in G.
(i) M contains at least one edge of every odd cycle C2k+1 in G, k ≥ 1, and exactly one
edge of every odd cycle C3 , C5 , C7 of G. Moreover, M contains every mid-edge
of any diamond in G.
(ii) No edge of any C4 can be in M.
(iii) If C is a C6 then either exactly two or none of the C-edges are in M.
159
Since every triangle contains exactly one M-edge and no M-edge is in any C4 , and the
pairwise distance of edges in any e.e.d. is at least 2, we obtain:
Corollary 12.3 If a graph G has an e.e.d. then G is K4 -free, gem-free and Ck -free for
any k ≥ 6.
As a consequence of K4 -freeness, chordal graphs with e.e.d. have bounded treewidth.
In [346], it was shown that the EED problem is solvable in linear time on chordal
graphs. This also follows from bounded treewidth.
For dually chordal graphs, the following lemma is helpful:
Lemma 12.2 ([95]) If G is a graph with an e.e.d. then G is chordal if and only if G
is dually chordal.
Proof. ⇒: Let G = (V, E) be a chordal graph with e.e.d. M. Then G is gem-free and
thus, sun-free chordal, i.e., strongly chordal. It is well known that strongly chordal
graphs are dually chordal. Thus, G is dually chordal.
⇐: Let G = (V, E) be a dually chordal graph with e.e.d. M. Let σ = (v1 , . . . , vn ) be a
maximum neighborhood ordering of G. If G is not chordal then G contains a chordless
cycle C isomorphic to some Ck , k ≥ 4. Let w = vi be the leftmost vertex of C in σ;
w has a maximum neighbor u in Gi := G[{vi , . . . , vn }]. Note that w 6= u since C is
chordless. Now if C is a C4 then u is adjacent to all vertices of C and thus, G contains
a W4 , and if C is a Ck for k ≥ 5 then u is adjacent to all vertices of a P4 in C and thus
contains a gem which is a contradiction in both cases. Thus, G is chordal which shows
Lemma 12.2.
2
Corollary 12.4 The EED problem can be solved in linear time for dually chordal
graphs.
In [82], EED is solved in linear time for chordal bipartite graphs and in polynomial
time for hole-free graphs (and thus, by Corollary 12.3, for weakly chordal graphs).
In [332], a linear time algorithm for EED on circular-arc graphs is given.
Efficient edge dominating sets are closely related to maximum induced matchings; it
is not hard to see that every efficient edge dominating set is a maximum induced
matching but of course not vice versa. However, when the graph has an efficient edge
dominating set and is regular then every maximum induced matching is an efficient edge
dominating set [122]. On the other hand, the complexity of the two problems differs
on some classes such as claw-free graphs where the MIM problem is NP-complete [309]
(even on line graphs) while the EED problem is solvable in polynomial time [123].
While every graph has a maximum induced matching, this is not the case for efficient
edge dominating sets. Thus, if the graph G has an efficient edge dominating set, this
gives also a maximum induced matching but in the other case, the MIM problem is
hard for claw-free graphs.
160
12.7
Graph Isomorphism
The Graph Isomorphism problem was shown to be isomorphism-complete, that is as
hard as in the general case, for strongly chordal graphs and chordal bipartite graphs
[428]. The Graph Isomorphism problem for distance-hereditary graphs is solvable in
linear time [429] (a first step for this was done in [173]); see also [373].
161
13
Metric Tree-Like Structures in Graphs
There are few other graph parameters measuring tree likeness of a (unweighted) graph
from a metric point of view. Two of them are also based on the notion of treedecomposition of Robertson and Seymour [402] (see Definition 10.4).
13.1
Tree-Breadth, Tree-Length and Tree-Stretch of Graphs
The length of a tree-decomposition T of a graph G is λ := maxi∈I maxu,v∈Vi dG (u, v)
(i.e., each bag Vi has diameter at most λ in G). The tree-length of G, denoted by
tl(G), is the minimum of the length over all tree-decompositions of G [188]. As chordal
graphs are exactly those graphs that have a tree decomposition where every bag is
a clique [113, 228, 432], we can see that tree-length generalizes this characterization
and thus the chordal graphs are exactly the graphs with tree-length 1. Note that treelength and treewidth are not related to each other graph parameters. For instance,
a clique on n vertices has tree-length 1 and treewidth n − 1, whereas a cycle on 3n
vertices has treewidth 2 and tree-length n. One should also note that many graph
classes with unbounded treewidth have bounded tree-length, such as interval, split,
permutation, chordal, and even AT-free graphs [188]. Analyzing a number of real-life
networks, taken from different domains like Internet measurements, biological datasets,
web graphs, social and collaboration networks, performed in [366] and in [2] shows that
those networks have sufficiently large treewidth but their tree-length is relatively small.
The breadth of a tree-decomposition T of a graph G is the minimum integer r such
that for every i ∈ I there is a vertex vi ∈ V with Vi ⊆ N r [vi ] (i.e., each bag Vi can be
covered by a disk N r [vi ] := {u ∈ V (G) : dG (u, vi ) ≤ r} of radius at most r in G). Note
that vertex vi does not need to belong to Vi . The tree-breadth of G, denoted by tb(G),
is the minimum of the breadth over all tree-decompositions of G [192]. Evidently, for
any graph G, 1 ≤ tb(G) ≤ tl(G) ≤ 2tb(G) holds. Hence, if one parameter is bounded
by a constant for a graph G then the other parameter is bounded for G as well.
Note that the notion of acyclic (R, D)-clustering of a graph introduced in [193] combines
tree-breadth and tree-length into one notion. Graphs admitting acyclic (D, D)-clustering are exactly graphs with tree-length at most D, and graphs admitting acyclic
(R, 2R)-clustering are exactly graphs with tree-breadth at most R. Hence, all chordal,
chordal bipartite, and dually chordal graphs have tree-breadth 1 [193].
In view of tree-decomposition T of G, the smaller parameters tl(G) and tb(G) of G
are, the closer graph G is to a tree metrically. Unfortunately, while graphs with treelength 1 (as they are exactly the chordal graphs) can be recognized in linear time,
the problem of determining whether a given graph has tree-length at most λ is NPcomplete for every fixed λ > 1 (see [336]). Judging from this result, it is conceivable
that the problem of determining whether a given graph has tree-breadth at most ρ is
NP-complete, too. 3-Approximation algorithms for computing the tree-length and the
tree-breadth of a graph are proposed in [188, 192, 2].
162
Proposition 13.1 ([188]) There is a linear time algorithm that produces for any
graph G a tree-decomposition of length at most 3tl(G) + 1.
Proposition 13.2 ([192, 2]) There is a linear time algorithm that produces for any
graph G a tree-decomposition of breadth at most 3tb(G).
It also follows from results of [141] and [188] that any graph G with small tree-length
or small tree-breadth can be embedded into a tree with a small additive distortion.
Proposition 13.3 For any (unweighted) connected graph G = (V, E) there is an unweighted tree H = (V, F ) (on the same vertex set but not necessarily a spanning tree
of G) for which the following is true:
∀u, v ∈ V, dH (u, v) − 2 ≤ dG (u, v) ≤ dH (u, v) + 3 tl(G) ≤ dH (u, v) + 6 tb(G).
Such a tree H can be constructed in O(|E|) time.
Previously, this type of results was known for chordal graphs and dually chordal graphs
[61], k-chordal graphs [138] and δ-hyperbolic graphs [139].
Graphs with small tree-length or small tree-breadth have many other nice properties. Every n-vertex graph with tree-length tl(G) = λ has an additive 2λ-spanner
with O(λn + n log n) edges and an additive 4λ-spanner with O(λn) edges, both constructible in polynomial time [187]. Every n-vertex graph G with tb(G) = ρ has a
system of at most log2 n collective additive tree (2ρ log2 n)-spanners constructible in
polynomial time [191]. Those graphs also enjoy a 6λ-additive routing labeling scheme
with O(λ log2 n) bit labels and O(log λ) time routing protocol [186], and a (2ρ log2 n)additive routing labeling scheme with O(log3 n) bit labels and O(1) time routing protocol with O(log n) message initiation time (by combining results of [191] and [196]).
See appropriate papers for more details.
Here we elaborate a bit more on a connection established in [192] between the treebreadth and the tree-stretch of a graph (and the corresponding tree t-spanner problem).
The tree-stretch ts(G) of a graph G = (V, E) is the smallest number t such that G
admits a spanning tree T = (V, E ′ ) with dT (u, v) ≤ tdG (u, v) for every u, v ∈ V. T is
called a tree t-spanner of G and the problem of finding such tree T for G is known
as the tree t-spanner problem. Note that as T is a spanning tree of G, necessarily
dG (u, v) ≤ dT (u, v) and E ′ ⊆ E. It is known that the tree t-spanner problem is NPhard [118]. The best known approximation algorithms have approximation ratio of
O(log n) [201, 192].
The following two results were obtained in [192].
Proposition 13.4 ([192]) For every graph G, tb(G) ≤ ⌈ts(G)/2⌉ and tl(G) ≤ ts(G).
Proposition 13.5 ([192]) For every n-vertex graph G, ts(G) ≤ 2tb(G) log2 n. Furthermore, a spanning tree T of G with dT (u, v) ≤ (2tb(G) log2 n) dG (u, v), for every
u, v ∈ V, can be constructed in polynomial time.
163
Proposition 13.5 is obtained by showing that every n-vertex graph G with tb(G) =
ρ admits a tree (2ρ log2 n)-spanner constructible in polynomial time. Together with
Proposition 13.4, this provides a log2 n-approximate solution for the tree t-spanner
problem in general unweighted graphs.
13.2
Hyperbolicity of Graphs and Embedding Into Trees
δ-Hyperbolic metric spaces have been defined by M. Gromov [250] in 1987 via a simple
4-point condition: for any four points u, v, w, x, the two larger of the distance sums
d(u, v) + d(w, x), d(u, w) + d(v, x), d(u, x) + d(v, w) differ by at most 2δ. They play
an important role in geometric group theory, geometry of negatively curved spaces,
and have recently become of interest in several domains of computer science, including
algorithms and networking. For example,
(a) it has been shown empirically in [409] (see also [1]) that the Internet topology
embeds with better accuracy into a hyperbolic space than into an Euclidean space
of comparable dimension,
(b) every connected finite graph has an embedding in the hyperbolic plane so that the
greedy routing based on the virtual coordinates obtained from this embedding is
guaranteed to work (see [305]).
A connected graph G = (V, E) equipped with standard graph metric dG is δ-hyperbolic if
the metric space (V, dG ) is δ-hyperbolic. More formally, let G be a graph and u, v, w and
x be its four vertices. Denote by S1 , S2 , S3 the three distance sums, dG (u, v) + dG (w, x),
dG (u, w) + dG (v, x) and dG (u, x) + dG (v, w) sorted in non-decreasing order S1 ≤ S2 ≤
2
S3 . Define the hyperbolicity of a quadruplet u, v, w, x as δ(u, v, w, x) = S3 −S
. Then
2
the hyperbolicity δ(G) of a graph G is the maximum hyperbolicity over all possible
quadruplets of G, i.e.,
δ(G) = max δ(u, v, w, x).
u,v,w,x∈V
δ-Hyperbolicity measures the local deviation of a metric from a tree metric; a metric
is a tree metric if and only if it has hyperbolicity 0. Note that chordal graphs have
hyperbolicity at most 1 [109], while k-chordal graphs have hyperbolicity at most k/4
[437].
The best known algorithm to calculate hyperbolicity has time complexity of O(n3.69 ),
where n is the number of vertices in the graph; it was proposed in [218] and involves
matrix multiplications. The authors of [218] also propose a 2-approximation algorithm
for calculating hyperbolicity that runs in O(n2.69 ) time and a 2 log2 n-approximation
algorithm that runs in O(n2 ) time.
According to [139], if a graph G has small hyperbolicity then it can be embedded to a
tree with a small additive distortion.
164
Proposition 13.6 ([139]) For any (unweighted) connected graph G = (V, E) with
n vertices there is an unweighted tree H = (V, F ) (on the same vertex set but not
necessarily a spanning tree of G) for which the following is true:
∀u, v ∈ V, dH (u, v) − 2 ≤ dG (u, v) ≤ dH (u, v) + O(δ(G) log n).
Such a tree H can be constructed in O(|E|) time.
Thus, the distances in n-vertex δ-hyperbolic graphs can efficiently be approximated
within an additive error of O(δ log n) by a tree metric and this approximation is sharp
(see [250] and [139, 227]). An earlier result of Gromov [250] established similar distance
approximations, however Gromov’s tree is weighted, may have Steiner points and needs
O(n2 ) time for construction.
It is easy to show that every graph G admitting a tree T with dG (x, y) ≤ dT (x, y) ≤
dG (x, y) + r for any x, y ∈ V is r-hyperbolic. Thus, the hyperbolicity of a graph G
indicates how G could be embedded into a tree with an additive distortion.
Graphs and general geodesic spaces with small hyperbolicities have many other algorithmic advantages. They allow efficient approximate solutions for a number of
optimization problems. For example, Krauthgamer and Lee [315] presented a PTAS
for the Traveling Salesman Problem when the set of cities lie in a hyperbolic metric space. Chepoi and Estellon [142] established a relationship between the minimum
number of balls of radius r + 2δ covering a finite subset S of a δ-hyperbolic geodesic
space and the size of the maximum r-packing of S and showed how to compute such
coverings and packings in polynomial time. In [139], Chepoi et al. gave efficient algorithms for fast and accurate estimations of diameters and radii of δ-hyperbolic geodesic
spaces and graphs. Additionally, Chepoi et al. showed in [140] that every n-vertex δhyperbolic graph has an additive O(δ log n)-spanner with at most O(δn) edges and
enjoys an O(δ log n)-additive routing labeling scheme with O(δ log2 n) bit labels and
O(log δ) time routing protocol.
The following relations between the tree-length and the hyperbolicity of a graph were
established in [139].
Proposition 13.7 ([139]) For every n-vertex graph G,
O(δ(G) log n).
δ(G)
≤
tl(G)
≤
Combining this with results from [192] (see Proposition 13.4 and Proposition 13.5),
one gets the following inequalities.
Proposition 13.8 ([190]) For any n-vertex graph G, δ(G) ≤ ts(G) ≤ O(δ(G) log2 n).
This proposition says, in particular, that every δ-hyperbolic graph G admits a tree
O(δ log2 n)-spanner. Furthermore, such a spanning tree for a δ-hyperbolic graph can
be constructed in polynomial time (see [192]).
165
The problem of approximating a given graph metric by a “simpler” metric is well motivated from several different perspectives. A particularly simple metric of choice, also
favored from the algorithmic point of view, is a tree metric, i.e., a metric arising from
shortest path distance on a tree containing the given points. In recent years, various
authors considered problems of minimum distortion embeddings of graphs into trees
(see [4, 19, 20, 141]), most popular among them being a non-contractive embedding
with minimum multiplicative distortion.
Let G = (V, E) be a graph. The (multiplicative) tree-distortion td(G) of G is the
smallest number α such that G admits a tree (not necessarily a spanning tree, possibly
weighted and with Steiner points) with
∀u, v ∈ V, dG (u, v) ≤ dT (u, v) ≤ α dG (u, v).
The problem of finding, for a given graph G, a tree T = (V ∪ S, F ) satisfying
dG (u, v) ≤ dT (u, v) ≤ td(G)dG (u, v), for all u, v ∈ V , is known as the problem of minimum distortion non-contractive embedding of graphs into trees. In a non-contractive
embedding, the distance in the tree must always be larger that or equal to the distance
in the graph, i.e., the tree distances “dominate” the graph distances.
It is known that this problem is NP-hard, and even more, the hardness result of [4]
implies that it is NP-hard to approximate td(G) better than γ, for some small constant
γ. The best known 6-approximation algorithm using layering partition technique was
recently given in [141]. It improves the previously known 100-approximation algorithm
from [20] and 27-approximation algorithm from [19].
The following interesting result was presented in [141].
Proposition 13.9 ([141]) For any (unweighted) connected graph G = (V, E) with
n vertices there is an unweighted tree H = (V, F ) (on the same vertex set but not
necessarily a spanning tree of G) for which the following is true:
∀u, v ∈ V, dH (u, v) − 2 ≤ dG (u, v) ≤ dH (u, v) + 3 td(G).
Such a tree H can be constructed in O(|E|) time.
Surprisingly, a multiplicative distortion is turned into an additive one. Moreover,
while a tree T = (V ∪ S, F ) satisfying dG (u, v) ≤ dT (u, v) ≤ td(G)dG (u, v), for all
u, v ∈ V , is NP-hard to find, tree H of Proposition 13.9 is constructible in O(|E|) time.
Furthermore, H is unweighted and has no Steiner points.
By adding at most n = |V | new Steiner points to tree H and assigning proper weights
to edges of H, the authors of [141] achieve a good non-contractive embedding of a
graph G into a tree.
Proposition 13.10 ([141]) For any (unweighted) connected graph G = (V, E) there
is a weighted tree Hℓ′ = (V ∪ S, F ) for which the following is true:
∀u, v ∈ V, dG (x, y) ≤ dHℓ′ (x, y) ≤ 3td(G)(dG (x, y) + 1).
Such a tree Hℓ′ can be constructed in O(|V ||E|) time.
166
As pointed out in [141], tree Hℓ′ provides a 6-approximate solution to the problem of
minimum distortion non-contractive embedding of an unweighted graph into a tree.
We conclude this section with one more chain of inequalities establishing relations
between the tree-stretch, the tree-length and the tree-distortion of a graph.
Proposition 13.11 ([190]) For every n-vertex graph G, tl(G) ≤ td(G) ≤ ts(G) ≤
2td(G) log2 n.
Proposition 13.11 says that if a graph G is non-contractively embeddable into a tree
with distortion td(G) then it is embeddable into a spanning tree with stretch at most
2td(G) log2 n. Furthermore, a spanning tree with stretch at most 2td(G) log2 n can be
constructed for G in polynomial time.
167
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