G - Isaac Newton Institute

On (k,l)-Leaf Powers
Peter Wagner
University of Rostock, Germany
(joint work with A. Brandstädt,
C. Hundt, V. B. Le and R. Sritharan)
Phylogenetic Trees
[Y. Kim, T. Warnow, Tutorial on Phylogenetic
Tree Estimation, 1999]:
The genealogical history of life (also called
evolutionary tree or phylogenetic tree) is
usually represented by a bifurcating, leaflabelled tree (i.e., leaves are labelled by the
species).
Phylogenetic Trees
The phylogenetic tree is rooted at the most
recent common ancestor of a set of taxa
(species, biomolecular sequences, languages
etc.), and each internal node is labelled by a
(hypothesised or known) ancestor.
Phylogenetic Roots and Powers
[Lin, Kearney, Jiang, Phylogenetic k-root and
Steiner k-root, ISAAC 2000]:
Let G = (V,E) be a finite undirected graph.
A tree T with leaf set V is a phylogenetic k-root
of G if the internal nodes of T have degree 3
and, for all leaves x and y,
xy  E  distT (x,y)  k.
Phylogenetic Roots and Powers
This notion is based on the idea that sequences
with a small (respectively, large) Hamming
distance should correspond to leaves with a
small (respectively, large) distance (number of
edges of the unique path connecting them) in a
phylogenetic tree.
Phylogenetic Roots and Powers
Arising problems:
Given a graph G, is there a phylogenetic k-root
of G? (k fixed)
[Lin, Kearney, Jiang, 2000]:
Linear time algorithm for k  4; open for k  5.
Phylogenetic Roots and Powers
Arising problems:
Given a graph G, is there a phylogenetic k-root
of G? (k fixed)
[Lin, Kearney, Jiang, 2000]:
Linear time algorithm for k  4; open for k  5.
Variant where vertices of V might appear as
internal nodes of T: Steiner k-root of G.
Leaf Powers
[Nishimura, Ragde, Thilikos, On graph powers
for leaf-labeled trees, J. Algorithms 2002]:
A finite undirected graph G = (V,E) is a k-leaf
power if there is a tree T = (U, F ) (called a kleaf root of G) with leaf set V, such that, for all
x,y  V,
xy  E  distT (x,y)  k.
Leaf Powers
A finite undirected graph G = (V,E) is a leaf
power if it is a k-leaf power, for some k  2.
So the leaf powers are, as T runs through all
trees, the subgraphs induced by the leaf set of T
of the various powers of T.
Obviously, the 2-leaf powers are exactly the
disjoint unions of cliques.
Leaf Powers
1
2
3
4
Leaf Powers
1
2
3
4
1
2
3
4
Leaf Powers
1
2
3
4
1
2
3
4
1
2
3
4
Leaf Powers
2
1
3
4
Leaf Powers
2
1
3
4
1 2 3 4
Chordal Graphs
Graph G is chordal if it contains no chordless
cycles of length at least 4.
Chordal Graphs
Graph G is chordal if it contains no chordless
cycles of length at least 4.
Chordal graphs have many facets:
- clique separators
- clique tree
- simplicial elimination orderings
- intersection graphs of subtrees of a tree ...
Graph Powers
For graph G = (V,E), let Gk = (V, Ek) with
xy  Ek  distG (x,y)  k
denote the k-th power of G.
Fact. A k-leaf power is an induced subgraph of
the k-th power of a tree, and every induced
subgraph of a k-leaf power is a k-leaf power.
Fact. Powers of trees are chordal.
Leaf Powers
A graph is strongly chordal if it is chordal and
sun-free. Trees are strongly chordal.
Theorem [Lubiw 1982; Dahlhaus, Duchet
1987; Raychaudhuri 1992] For every k  2:
G strongly chordal  Gk strongly chordal.
Corollary. For every k  2, k-leaf powers are
strongly chordal.
Leaf Powers
[Bibelnieks, Dearing, Neighborhood subtree
tolerance graphs, 1993], based on
[Broin, Lowe, A dynamic programming
algorithm for covering problems with (greedy)
totally balanced constraint matrices, 1986]:
Fact. There is a strongly chordal graph which
is not a k-leaf power, for any k.
Not a Leaf Power
3- and 4-Leaf Powers
[Nishimura, Ragde, Thilikos, 2002]:
(Very complicated) O(n3) algorithms for
recognising 3- and 4-leaf powers
3- and 4-Leaf Powers
[Nishimura, Ragde, Thilikos, 2002]:
(Very complicated) O(n3) algorithms for
recognising 3- and 4-leaf powers
Open:
- Characterisation of k-leaf powers, for k  5,
and
- Characterisation of leaf powers in general.
Leaf Powers
[Lin, Kearney, Jiang 2000]
A critical clique of G is a maximal clique
module in G.
The critical clique graph cc(G) of G is the
graph whose vertices are the critical cliques of
G, and two such cliques are adjacent iff they
contain vertices adjacent in G.
8
3
1
11
13
4
2
5
6
7
12
9
10
8
3
1
11
13
4
2
5
6
7
12
9
10
8
3
1
11
13
4
2
5
6
12
9
10
7
3,4
1,2
8,9
11,12
13
5,6
7
10
3-Leaf Powers
3-Leaf Powers
3-Leaf Powers
3-Leaf Powers
Theorem [Dom, Guo, Hüffner, Niedermeier
2004]
G is a 3-leaf power  G is (bull, dart, gem)-free
chordal  cc(G) is a tree.
3-Leaf Powers
[Brandstädt, Le 2005; Rautenbach 2004]
A connected graph G is a 3-leaf power  G is
the result of substituting cliques into the
vertices of a tree.
[Brandstädt, Le 2005]
Linear time recognition for 3-leaf powers.
4-Leaf Powers
G2
G1
G5
G6
G3
G4
G7
G8
4-Leaf Powers
Theorem [Rautenbach 2004]
A graph G without true twins (basic) is a 4-leaf
power  G is (G1, ..., G8)-free chordal.
4-Leaf Powers
Theorem [Brandstädt, Le, Sritharan 2005]
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 is chordal, 2-connected and (G1, ..., G5)free.
4-Leaf Powers
Theorem [Brandstädt, Le, Sritharan 2005]
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 (G1, ..., G8)-free chordal.
More results on Leaf Powers
Theorem [Brandstädt, Hundt 2007]
Ptolemaic (i.e. distance-hereditary chordal, i.e.
gem-free chordal) graphs and interval graphs
are leaf powers.
This is implied by the later result:
Theorem [Wagner 2007]
Rooted directed path graphs are leaf powers.
More results on Leaf Powers
Clearly, for any given k, every k-leaf power is a
(k+2)-leaf power (subdivide leaf edges).
But what about k- and (k+1)-leaf powers?
2- are 3-leaf powers, and 3- are 4-leaf powers.
Theorem [Brandstädt, Wagner 2007]
For all k>3, there is a k-leaf power which is not
a (k+1)-leaf power.
(k,l)-Leaf Powers
A finite undirected graph G = (V,E) is a (k,l)leaf power if there is a tree T = (U, F ) with leaf
set V, such that, for all x,y  V,
xy  E  distT (x,y)  k, and
xy  E  distT (x,y)  l.
Such a tree T is a (k,l)-leaf root of G.
Clearly, k-leaf powers are (k,k+1)-leaf powers.
(4,6)-Leaf Powers
Theorem [Brandstädt, Wagner 2007]
For a connected graph G, the following are equivalent:
(i) G is a (4,6)-leaf power,
(ii) G is strictly chordal, i.e., (dart,gem)-free chordal,
(iii) G results from a block graph by substituting
cliques into its vertices.
(A paper by Kennedy, Lin and Yan 2006 shows that strictly
chordal graphs are leaf powers.)
(k,l)-Leaf Powers
As the 2- and 3-leaf powers, the strictly chordal
graphs ((4,6)-leaf powers) are the class of (k,l)leaf powers, for infinitely many pairs (k,l).
Characterisations are also known for (6,8)- and
(8,11)-leaf powers.
G1
G2
G4
G7
G3
G5
G8
G6
G9
(6,8)-Leaf Powers
Theorem [Brandstädt, Wagner 2007]
The following conditions are equivalent:
(i) G is a basic (6,8)-leaf power.
(ii) G is an induced subgraph of the square of
some block graph.
(iii) G is (G1, ..., G9)-free chordal.
2,3
3,4
3,5
4,5
4,6
4,7
5,6
5,7
5,8
5,9
6,7
6,8
6,9
6,10 6,11
7,8
7,9
7,10 7,11 7,12 7,13
8,9
8,10 8,11 8,12 8,13 8,14 8,15
9,10 9,11 9,12 9,13 9,14 9,15 9,16
10,11 10,12 10,13 10,14 10,15 10,16
11,12 11,13 11,14 11,15 11,16
12,13 12,14 12,15 12,16
13,14 13,15 13,16
14,15 14,16
15,16
(k,l)-Leaf Powers
Open Problems:
1. Characterisation of k-leaf powers, for k  5,
and of leaf powers in general.
2. Complexity of recognising k-leaf powers,
for k  6, and of leaf powers in general.
3. Characterisation of further (k,l)-leaf powers,
e.g. for (k,l)=(7,9) or (k,l)=(8,10).
Thank you for your attention!

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