On (k,l)-Leaf Powers

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On (k,l)-Leaf Powers

• Peter Wagner
• University of Rostock, Germany
• (joint work with A. Brandstädt,
• C. Hundt, V. B. Le and R. Sritharan)

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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, leaf-labelled tree (i.e., leaves are
labelled by the species).
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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.
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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.
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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.
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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.
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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.
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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 k-leaf root
of G) with leaf set V, such that, for all x,y ?
V, xy ? E ? distT (x,y) ? k.
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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.
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Leaf Powers
2
4
1
3
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Leaf Powers
2
4
1
3
2
4
1
3
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Leaf Powers
2
4
1
3
2
4
1
3
2
4
1
3
13
Leaf Powers
2
4
1
3
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Leaf Powers
2
4
1
3
2
4
1
3
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Chordal Graphs
Graph G is chordal if it contains no chordless
cycles of length at least 4.
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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 ...

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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.
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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.
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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.
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Not a Leaf Power
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3- and 4-Leaf Powers
Nishimura, Ragde, Thilikos, 2002 (Very
complicated) O(n3) algorithms for recognising 3-
and 4-leaf powers
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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.
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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.
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3
8
11

1
13
4
2
12
9
5
6
10
7
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3
8
11

1
13
4
2
12
9
5
6
10
7
27
3
8
11

1
13
4
2
12
9
5
6
10
7
3,4
8,9
11,12
1,2
13
5,6
10
7
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3-Leaf Powers
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3-Leaf Powers
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3-Leaf Powers
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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.
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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.
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4-Leaf Powers
G2
G3
G4
G1
G5
G6
G7
G8
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4-Leaf Powers
Theorem Rautenbach 2004 A graph G without true
twins (basic) is a 4-leaf power ? G is (G1, ...,
G8)-free chordal.
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4-Leaf Powers

• Theorem Brandstädt, Le, Sritharan 2005
• For every graph G, the following conditions are
  equivalent
• G is a 2-connected basic 4-leaf power.
• G is the square of some tree.
• G is chordal, 2-connected and (G1, ..., G5)-free.

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4-Leaf Powers

• Theorem Brandstädt, Le, Sritharan 2005
• The following conditions are equivalent
• G is a basic 4-leaf power.
• 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.
• G is an induced subgraph of the square of some
  tree.
• G is (G1, ..., G8)-free chordal.

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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.
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More results on Leaf Powers
Clearly, for any given k, every k-leaf power is a
(k2)-leaf power (subdivide leaf edges). But what
about k- and (k1)-leaf powers? 2- are 3-leaf
powers, and 3- are 4-leaf powers. Theorem
Brandstädt, Wagner 2007 For all kgt3, there is a
k-leaf power which is not a (k1)-leaf power.
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(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,k1)-leaf powers.
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(4,6)-Leaf Powers

• Theorem Brandstädt, Wagner 2007
• For a connected graph G, the following are
  equivalent
• G is a (4,6)-leaf power,
• G is strictly chordal, i.e., (dart,gem)-free
  chordal,
• 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.)

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(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.
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G2
G1
G3
G6
G4
G5
G9
G8
G7
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(6,8)-Leaf Powers

• Theorem Brandstädt, Wagner 2007
• The following conditions are equivalent
• G is a basic (6,8)-leaf power.
• G is an induced subgraph of the square of some
  block graph.
• G is (G1, ..., G9)-free chordal.

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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
8,13
8,14
8,15
8,12
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
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(k,l)-Leaf Powers

• Open Problems
• Characterisation of k-leaf powers, for k ? 5, and
  of leaf powers in general.
• Complexity of recognising k-leaf powers, for k ?
  6, and of leaf powers in general.
• Characterisation of further (k,l)-leaf powers,
  e.g. for (k,l)(7,9) or (k,l)(8,10).

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Thank you for your attention!