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, leaf-labelled 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 k-leaf 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 4 3

Leaf Powers 2 1 4 3 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.

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

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

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

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 G1 G2 G3 G4 G5 G6 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: 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.

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.

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: 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.)

(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 G3 G4 G5 G6 G7 G8 G9

(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.

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: 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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