On Leaf Powers Andreas Brandstädt University of Rostock, Germany (joint work with Van Bang Le, Peter Wagner, Christian Hundt, 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- labeled tree (i.e., leafs are labeled 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 the internal nodes of the tree are each labeled by a (hypothesized 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 xy  E  dist T (x,y)  k.

Phylogenetic Roots and Powers Corresponding problems: PRk: Given a graph G, is there a phylogenetic k-root of G? (k fixed) [Lin, Kearney, Jiang, 2000]: Linear time for k  4; open for k  5.

Phylogenetic Roots and Powers Corresponding problems: PRk: Given a graph G, is there a phylogenetic k-root of G? (k fixed) [Lin, Kearney, Jiang, 2000]: Linear time 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 ) with leaf set V such that for all x,y  V xy  E  dist T (x,y)  k. Such a tree T is a k-leaf root of G.

Leaf Powers A finite undirected graph G = (V,E) is a leaf power if it is a k-leaf power for some k  2. Obviously, the 2-leaf powers are exactly the disjoint unions of cliques.

Leaf Powers

Chordal Graphs Graph G is chordal if it contains no chordless cycles of length at least four.

Chordal Graphs Graph G is chordal if it contains no chordless cycles of length at least four. 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 G k = (V, E k ) with xy  E k  dist G (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  G k 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 are strongly chordal graphs which are no k-leaf power for any k.

No Leaf Power

3- and 4-Leaf Powers [Nishimura, Ragde, Thilikos, 2002]: (Very complicated) O(n 3 ) algorithms for recognizing 3- and 4-leaf powers

3- and 4-Leaf Powers [Nishimura, Ragde, Thilikos, 2002]: (Very complicated) O(n 3 ) algorithms for recognizing 3- and 4-leaf powers Open: - Characterization of k-leaf powers for k  5 and - Characterization 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.

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

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 [B., 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. [B., Le 2005] Linear time recognition for 3-leaf powers.

4-Leaf Powers G1G1 G2G2 G3G3 G4G4 G5G5 G6G6 G7G7 G8G8

Theorem [Rautenbach 2004] A graph G without true twins is a 4-leaf power  G is (G 1,..., G 8 )-free chordal.

4-Leaf Powers Theorem [B., 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 (G 1,..., G 5 )- free.

4-Leaf Powers Theorem [B., 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 (G 1,..., G 8 )-free chordal.

(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  dist T (x,y)  k and xy  E  dist T (x,y)  l. Such a tree T is a k-leaf root of G.

(4,6)-Leaf Powers Theorem [B., Wagner 2007] For 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. (There is a paper on strictly chordal graphs by Kennedy, Lin and Yan 2006 showing that these graphs are leaf powers.)

2,3 3,43,5 4,54,64,7 5,65,75,85,9 6,76,86,96,106,11 7,87,97,107,117,127,13 8,98,108,118,138,148,15 9,109,119,129,139,149,159,16 10,1110,1210,1310,1410,1510,16 11,1211,1311,1411,1511,16 12,1312,1412,1512,16 13,1413,1513,16 14,1514,16 15,16 8,12

Ptolemaic Graphs are Leaf Powers Theorem [B., Hundt 2007] Every ptolemaic graph, i.e., gem-free chordal graph is a k-leaf power for some k.

Ptolemaic Graphs are Leaf Powers For a leaf power G, let l(G) denote the smallest k such that G is a k-leaf power. We call l(G) the leaf rank of G. Theorem [B., Hundt 2007] Ptolemaic graphs have unbounded leaf rank.

Leaf Powers Open Problems: 1.Characterization of k-leaf powers for k  5 and of leaf powers in general. 2.Complexity of recognizing k-leaf powers for k  6 and of leaf powers in general. 3.Is every k-leaf power also a (k+1)-leaf power?

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