On Leaf Powers
Andreas Brandstädt
Lehrstuhl für Theoretische Informatik, Institut für Informatik
Universität Rostock, D-18051 Rostock, Germany.
[email protected]
Abstract
For an integer k, a tree T is a k-leaf root of a finite simple undirected graph G = (V, E)
if the set of leaves of T is the vertex set V of G and for any two vertices x, y ∈ V , x 6= y,
xy ∈ E if and only if the distance of x and y in T is at most k. Then graph G is a k-leaf
power if it has a k-leaf root. G is a leaf power if it is a k-leaf power for some k. This notion
was introduced and studied by Nishimura, Ragde and Thilikos; it has its background and
motivation in computational biology and phylogeny.
In this survey, we describe recent results on leaf powers, variants and generalizations.
We discuss the relationship between leaf powers and strongly chordal graphs as well as
fixed tolerance NeST graphs, describe some subclasses of leaf powers, give the complete
inclusion structure of k-leaf power classes, and describe various characterizations of 3and 4-leaf powers, as well as of distance-hereditary 5-leaf powers. Finally we discuss two
variants of the notion of k-leaf power such as (k, ℓ)-leaf powers and exact leaf powers,
and we generalize leaf powers (of trees) to simplicial powers of graphs. Most of the
presented results are part of joint work, mostly with Van Bang Le and Peter Wagner,
but also with Christian Hundt, Federico Mancini, R. Sritharan, and Dieter Rautenbach.
Keywords and Classification: Leaf powers; leaf roots; phylogenetic trees; strongly chordal
graphs; graph powers; tree powers; exact leaf powers; simplicial powers; linear time
recognition; characterization by forbidden induced subgraphs.
1
Introduction
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 [20, 45].
Motivated by this background, Nishimura, Ragde and Thilikos [52] defined the following
notions: A tree T is a k-leaf root of a finite undirected graph G = (V, E) if the set of leaves
of T is V 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 [45] 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.
1
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 paper is to give a survey on some recent results on leaf powers. In particular,
we report on work on leaf powers recently done in the group of Theoretical Computer Science
at the University of Rostock [5, 6, 7, 8, 9, 10, 11, 13, 14, 64]. A corresponding talk was given
by the author at the CanaDAM 2009 conference in Montreal in May 2009.
This paper is organized as follows:
− In section 2, we present some basic notions and results.
− In section 3, we collect some basic facts on leaf powers. In particular, we mention that
leaf powers are strongly chordal but not vice versa, a graph is a leaf power if and only
if it is a fixed tolerance NeST graph, and we discuss some interesting subclasses of leaf
powers such as (unit) interval graphs and rooted directed path graphs.
− In section 4 we describe the inclusion structure of k-leaf power classes.
− In section 5 we describe characterizations of 3-leaf powers, 4-leaf powers and distancehereditary 5-leaf powers as well as new characterizations of squares of trees.
− Finally, in section 6 we describe results on three variants of leaf powers, namely on
(k, ℓ)-leaf powers, on exact leaf powers and on simplicial powers; the last one represents
a natural generalization of leaf powers.
Support by German Research Council (Deutsche Forschungsgemeinschaft) DFG BR 2479/7-1
is gratefully acknowledged.
2
Basic notions and results
Throughout this paper, 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. 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. A stable set is a set of vertices
which are mutually nonadjacent.
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 paper, 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 1-connected components of G are the
2
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 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 2) has
five vertices such that four of them induce a P4 and the fifth is adjacent to all of them.
A graph is chordal if it contains no induced Ck , k ≥ 4. A graph is strongly chordal if it
is chordal and sun-free [26] (see also [12] for various characterizations of (strongly) chordal
graphs). Graph G is distance hereditary if for all connected induced subgraphs G′ of G, the
distance function in G′ is the same as the restriction of the distance function of G to V (G′ ). It
is well known that a chordal graph is distance hereditary if and only if it is gem-free ([34, 36],
see also [12]). These graphs are also called ptolemaic graphs. A connected graph G is a block
graph if its its 2-connected components (i.e., its blocks) are cliques. It is well known that a
connected graph is a block graph if and only if it is diamond-free and chordal (see e.g. [1]).
Note that there is a close connection between these graph classes and certain acyclicity
conditions of hypergraphs motivated from relational database schemes which are described
by Fagin [25]: Let C(G) denote the hypergraph of maximal cliques of graph G. Then
− G is a block graph if and only if C(G) is Berge-acyclic.
− G is distance-hereditary chordal if and only if C(G) is γ-acyclic.
− G is strongly chordal if and only if C(G) is β-acyclic.
− G is chordal if and only if C(G) is α-acyclic.
A bipartite graph is chordal bipartite if it contains no incuced cycle of length at least 6.
Interval graphs are the intersection graphs of intervals on the real line. Unit interval graphs
are interval graphs with unit length intervals. A graph is a split graph if its vertex set can be
partitioned into a clique and a stable set. Note that G is a split graph if and only if G and
its complement graph is chordal. See [12] for more information on all these graph classes.
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
3
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 , reDisjoint vertex sets X, Y form a join (cojoin, respectively), denoted by X Y
(X Y
spectively), if for all pairs x ∈ X, y ∈ Y , xy ∈ E (xy 6∈ E, respectively) holds.
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 5.1).
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 2.1 is a well-known fact;
see e.g. [55] by Roberts on indifference graphs.
Proposition 2.1 The inclusion-maximal clique modules of a graph are pairwise disjoint.
In [45], 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.
A linear time algorithm for constructing the critical clique graph CC(G) for a given chordal
graph G was already claimed in [45]; such an algorithm is given e.g. in [3, 48] where the
maximal clique modules of a (not necessarily chordal) graph are constructed in linear time.
For a linear time algorithm for modular decomposition see [49].
3
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.
3.1
Leaf powers are strongly chordal
In [22, 47, 54], it is shown that the class of strongly chordal graphs is closed under powers:
Theorem 3.1 ([22, 47, 54]) If G is strongly chordal then for every k ≥ 1, Gk is strongly
chordal.
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 3.1 (see also [38]):
Proposition 3.1 For every k ≥ 2, k-leaf powers are strongly chordal.
4
This strengthens the observation that k-leaf powers are chordal (see e.g. [24, 52, 53]).
Proposition 3.2 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.
3.2
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 [4] and by Hayward, Kearney and Malton [32, 33]. 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 [32, 33]:
Theorem 3.2 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 3.2, instead of using positive weights ω in
trees, in [4, 32, 59], 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
[4], based on [15], it is shown:
Theorem 3.3 Fixed tolerance NeST graphs are strongly chordal, and there are strongly
chordal graphs which are no fixed tolerance NeST graph.
In [6], we show:
Theorem 3.4 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 3.3 and 3.4, it shows
that leaf powers are a proper subclass of strongly chordal graphs: See Figure 1 for an example
of such a graph.
Corollary 3.1 There are strongly chordal graphs which are no k-leaf power for any k ≥ 2.
Secondly, since interval graphs are fixed tolerance NeST graphs [33], it implies that interval
graphs are leaf powers which was shown in a direct way in [5]. We strengthen this inclusion
in [6] by showing:
Theorem 3.5 Every rooted directed path graph is a leaf power but not vice versa.
5
Figure 1: A strongly chordal graph which is no k-leaf power for any k
3.3
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 3.1 for an example.
For the next result we implicitly use the concept of clique-width which will not be defined
here (see e.g. [29] for details); Golumbic and Rotics [29] showed that unit interval graphs
have unbounded clique-width. By a result of Todinca [61] (see also [30]), 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 3.3 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 3.2 Leaf powers have unbounded clique-width.
A caterpillar T is a tree consisting of a path (the backbone of T ) and some leaves attached
to the backbone. In [5], 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:
Theorem 3.6 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.
(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:
6
k−2
Lemma 3.1 ([14, 64]) For all k and k′ with 2 ≤ k′ < k, the (k − 2)-th power P2k−3
of the
induced path P2k−3 with 2k − 3 vertices is a k-leaf power but no k′ -leaf power.
k−2
. For some upper and lower bounds on the leaf
This means that k is the leaf rank of P2k−3
rank of interval graphs and of ptolemaic graphs see [5, 6].
4
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.
[28]. In [14] we generalized this result to all k:
Theorem 4.1 For every k ≥ 4, L(k) 6⊂ L(k + 1).
In [64], we give a complete description of containments between k-leaf powers and k′ -leaf
powers:
Theorem 4.2 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 4.2 implies:
Corollary 4.1 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 4.2 (i) follows by Lemma 3.1. The proof of Theorem 4.2 (ii) is long and very
technical. Among others, it makes use of Buneman’s Four-Point Condition (∗) for distances
in connected graphs which 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)}.
Theorem 4.3 highlights the metric similarity between trees and block graphs in terms of
Buneman’s four-point condition.
Theorem 4.3 Let G be a connected graph.
(i) Buneman [16]: G is a tree if and only if G contains no triangles and satisfies (∗).
(ii) Howorka [35]: G is a block graph if and only if G satisfies (∗).
7
5
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 4-leaf powers
but also a new characterization of squares of trees and some other results concerning distancehereditary 5-leaf powers.
5.1
Basic leaf powers and substitution of cliques
The following notion from [11] 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.
Observation 5.1
(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 5.1 (ii) does not hold; cf. the comments after
Proposition 5.2.
Proposition 5.1 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 5.2 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 5.2 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 5.1 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.
5.2
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:
8
bull
dart
gem
Figure 2: bull, dart and gem have no 3-leaf root
Observation 5.2 Every tree (forest, respectively) is a 3-leaf power, and a 3-leaf root of it
can be determined in linear time.
Theorem 5.1 below collects various characterizations of 3-leaf powers from [7, 23, 53]; for bull,
dart and gem see Figure 2.
Theorem 5.1 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 [7, 53] for more structural details and other equivalent conditions, and in particular [7]
for a simple linear time recognition algorithm of 3-leaf powers.
5.3
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 [31, 57]. Efficient algorithms for recognizing squares of trees were studied in
[38, 42, 46]. Lin and Skiena [46], and Lau [42] give a linear time algorithm for recognizing
whether a given graph is the square of a tree. In [19], 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.
For tree T = (V, ET ) let Tx denote the star with center x in T , i.e., Tx = NT [x].
Observation 5.3 ([11]) Let G = T 2 for a tree T .
equivalent:
9
Then the following conditions are
(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 .
Observation 5.4 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 .
Observation 5.4 is also mentioned in [42].
For the graphs G1 , . . . , G5 mentioned in Theorem 5.2, see Figure 3.
G1
G2
G6
G3
G5
G4
G7
G8
Figure 3: Graphs G1 - G8
In [11], we have characterized the squares of trees as follows:
Theorem 5.2 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:
(a) Every pair of distinct maximal cliques has at most two vertices in common.
(b) Every 2-cut belongs to exactly two maximal cliques of G.
(c) Every pair of nondisjoint 2-cuts belongs to the same maximal clique.
(d) 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.
As testing chordality [56, 60] and determining a clique tree [58] can be done in linear time,
Theorem 5.2 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. [42, 46]). Note that the linear
time algorithm for recognizing squares of trees given in [46] builds the tree root incrementally,
by identifying the leaves and their parents of a possible root tree, and repeating the process
10
recursively, while in [42] 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 5.2 ([42, 57]) The tree roots of squares of trees are unique up to isomorphism.
5.4
Structure of (basic) 4-leaf powers
The following characterization of 4-leaf powers by Rautenbach [53] inspired the results in the
previous section on squares of trees, and in particular Theorem 5.2 (for G1 -G8 see Figure 3).
Theorem 5.3 ([53]) Let G be a graph without true twins. Then G is a 4-leaf-power if and
only if G is chordal and (G1 , . . . , G8 )-free.
Subsequently we study 4-leaf powers in more detail. By Proposition 5.1, 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 4leaf powers in general. In [11] we have characterized the 2-connected basic 4-leaf powers as
follows:
Theorem 5.4 For every graph G, the following conditions are equivalent:
(i) G is a 2-connected basic 4-leaf power.
(ii) G is the square of some tree.
(iii) G has a basic 4-leaf root without invisible vertices.
Theorem 5.2 together with Theorem 5.4 has some consequences.
Corollary 5.3 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 5.4 The graphs G1 , G2 , G3 , G4 , and G5 in Figure 3 are not basic 4-leaf powers.
Moreover, G1 and G4 are not 4-leaf powers at all.
Corollary 5.5 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 5.6 The graphs G6 , G7 , and G8 in Figure 3 are not basic 4-leaf powers. Moreover,
G6 is not a 4-leaf power at all.
Observation 5.5 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− .
11
In [11], we have characterized the basic 4-leaf powers as follows:
Theorem 5.5 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.
Corollary 5.7 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 5.3, 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 5.2). The equivalence (i) ⇔ (iv) in Theorem 5.5 extends the
characterization of 4-leaf powers without true twins in Theorem 5.3 to the larger class of
basic 4-leaf powers. Theorem 5.5 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 [11], we give a characterization of
4-leaf powers in general and a linear time recognition algorithm for them.
5.5
Structure of (basic) distance-hereditary 5-leaf powers
The results in this section are from [10]. Recall that a chordal graph is distance-hereditary
if and only if it is gem-free [34, 36].
(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 .
(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.
Two-connected distance-hereditary basic 5-leaf powers can be characterized as follows (for
the graphs F1 , . . . , F8 see Figure 4).
Theorem 5.6 ([10]) Let G be a 2-connected distance-hereditary graph. Then the following
statements are equivalent:
(i) G is a basic 5-leaf power.
12
F1
F2
F4
F6
F3
F5
F8
F7
Figure 4: Forbidden induced subgraphs F1 , . . . , F8 of 2-connected basic distance-hereditary
5-leaf powers
(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 [10]
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 5.7 ([10]) For any distance-hereditary graph G, the following statements are
equivalent:
(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 5.8 ([10]) 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.
13
It remains an open problem to characterize 5-leaf powers in general. The characterization of
distance-hereditary basic 5-leaf powers in Theorem 5.7 is a promising first step. Note that in
[18], a linear time recognition of 5-leaf powers is given.
6
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.
6.1
On (k, ℓ)-leaf powers
In [13], 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 [41] as those chordal graphs whose clique hypergraph is a socalled strict hypertree are exactly the (dart,gem)-free chordal graphs according to Corollary
2.2.2. in [39]:
Proposition 6.1 A graph is strictly chordal if and only if it is (dart, gem)-free chordal.
The subsequent Theorem 6.1 has been our motivation for defining and investigating the
notion of (k, ℓ)-leaf powers in [13].
Theorem 6.1 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.
The equivalence of conditions (ii) and (iii) in Theorem 6.1 was shown already in [8].
14
H1
H2
H3
H5
H4
H6
Figure 5: Forbidden subgraphs H1 , . . . , H6 .
The equivalence of (ii) and (iv) is implicitly mentioned in [41] – Lemma 2.4 of [41] 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 [17].
Corollary 6.1 Strictly chordal graphs are 4-leaf powers and 5-leaf powers ( and thus also
k-leaf powers for all k ≥ 4).
Corollary 6.1 is one of the main results (namely Theorem 4.1) in [41]. It has also been
mentioned in Theorem 2.5 of [41] that strictly chordal graphs can be recognized in linear
time. The proof in [41] is based on a linear time algorithm for constructing the critical clique
graph CC(G) for a given chordal graph G (see section 2).
By Theorem 6.1, the linear time recognition of strictly chordal graphs given in [41] can be
simplified in the following way: 1) construct CC(G); 2) check whether CC(G) is a block graph;
according to Theorem 6.1 (iv), this recognizes strictly chordal graphs.
In [13], we give another, conceptually very simple, linear time algorithm for recognizing
(4, 6)-leaf powers without constructing CC(G).
Corollary 6.2 (4, 6)-leaf powers (and thus also strictly chordal graphs) can be recognized in
linear time.
Inspired by Theorem 5.3 and Theorem 5.5, 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
3 for G1 , G4 , G6 and Figure 5 for H1 − H6 ):
Theorem 6.2 The following conditions are equivalent:
(i) G is a basic (6, 8)-leaf power.
15
(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.
In [13] we give many other properties of (k, ℓ)-leaf power classes and characterize various
other classes of this type.
6.2
Exact leaf powers
The results of this section are from [9]. 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 (for the forbidden subgraphs A and
domino see Figure 6). Note that exact leaf powers are not a special case of leaf powers.
house
A
domino
Figure 6: Forbidden subgraphs of exact 4-leaf powers
Theorem 6.3 ([9]) For a connected graph G, the following statements are equivalent:
(i) G is an exact 3-leaf power.
(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 6.3 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 6.
Theorem 6.4 ([9]) 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 6 as an induced subgraph.
16
(iii) G is obtained from a block graph by substituting vertices by stable sets.
Corollary 6.4 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.
The problem of characterizing exact k-leaf powers for k ≥ 5 is open.
6.3
Simplicial powers of graphs
In [8] 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 ⊆ VH is the set of all simplicial vertices in H and for all x, y ∈ VG , xy ∈ EG if and only
if 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 [52], leaf powers are exactly the simplicial powers of trees.
As Proposition 6.2 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 a 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.
Proposition 6.2 ([8]) Every nontrivial graph is
(i) the 2-simplicial power of a split graph, and
(ii) the 4-leaf power of a bipartite graph.
Corollary 6.5 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 6.2 (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 6.5 2-simplicial split graph root is NP-complete.
17
The following Theorem 6.6 was the main motivation for introducing the concept of simplicial
powers:
Theorem 6.6 For k ≥ 2, a graph is a k-leaf power if and only if it is a (k − 1)-simplicial
power of a claw-free block graph.
It is based on Theorem 8.5 in [31]:
Theorem 6.7 A graph is the line graph of a tree if and only if it is a claw-free block graph.
Recall that by Theorem 5.1, G is a 3-leaf power (of a tree) if and only if G is (bull,dart,gem)free chordal. Theorem 6.8 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 6.1.
Theorem 6.8 ([8]) 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 arises from a block graph by replacing vertices by cliques.
Basic 3-simplicial powers of block graphs can be characterized in the following way:
Theorem 6.9 ([8]) 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 6.2 in the context of (6, 8)-leaf
powers in a different way. In [8], we give many other results on simplicial powers of graphs.
7
Concluding remarks
In this survey we present recent structural and algorithmic results on leaf powers; these
graphs are strongly chordal but not vice versa. A graph is a leaf power if and only if it is
a fixed tolerance NeST graph, and (unit) interval graphs and rooted directed path graphs
are interesting subclasses of leaf powers. We describe the inclusion structure of k-leaf power
classes, give characterizations of 3-leaf powers, 4-leaf powers and distance-hereditary 5-leaf
powers as well as a new characterization of squares of trees. Finally we describe results on
two variants of leaf powers, namely on (k, ℓ)-leaf powers as well as on exact leaf powers, and
on simplicial powers as a natural generalization of leaf powers.
18
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 [18]. 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.
Moreover, the complexity of determining the leaf rank of a given leaf power is an open
problem.
In [63], it is shown that the isomorphism problem for strongly chordal graphs is 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 [50] 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. [27, 42, 43, 44, 51]. Perhaps, techniques developed in these papers might be useful for
solving the open problems on leaf powers.
Acknowledgement. The author would like to thank all his coauthors on the leaf power
project as well as Chı́nh Hoàng for numerous discussions and the support by Wilfrid Laurier
University, Waterloo, Ontario.
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21