On Leaf Powers. Abstract

Transcription

1 On Leaf Powers Andreas Brandstädt Lehrstuhl für Theoretische Informatik, Institut für Informatik Universität Rostock, D Rostock, Germany. 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 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 3- and 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

2 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 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 deg G (v) of a vertex v is the number of its neighbors, deg G (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

3 usual connected components, and the 2-connected components of G are also called blocks of G. For graphs G 1 = (V 1,E 1 ), G 2 = (V 2,E 2 ), the graph G 1 G 2 (G 1 G 2, respectively) has vertex set V 1 V 2 (V 1 V 2, respectively) and edge set E 1 E 2 (E 1 E 2, respectively). Let d G (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 G k = (V,E k ) with xy E k if and only if d G (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 P k denote a chordless path with k vertices and k 1 edges, and for k 4, let C k 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 K r,s ; K 1,3 is also called claw. Let S k denote the (complete) sun with 2k vertices u 1,...,u k and w 1,...,w k such that u 1,...,u k is a clique, w 1,...,w k is a stable set and for all i {1,...,k}, N(w i ) = {u i,u i+1 } (index arithmetic modulo k). The diamond has four vertices and exactly one pair of nonadjacent vertices, i.e., it is a K 4 minus one edge, denoted K4. In general, for k 4, a clique with k vertices minus an edge is denoted K k ; it is the (k 2)-th power of the induced path P k. The gem (see Figure 2) has five vertices such that four of them induce a P 4 and the fifth is adjacent to all of them. A graph is chordal if it contains no induced C k, 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

4 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. Disjoint vertex sets X,Y form a join (cojoin, respectively), denoted by X 1 Y (X 0 Y, respectively), if for all pairs x X, y Y, xy E (xy 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 N G (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, G k 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

5 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 d T (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 d T (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

6 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. The following powers P k 2 2k 3 of induced paths P 2k 3 with 2k 3 vertices are unit interval graphs for which the leaf rank can be determined: 6

7 Lemma 3.1 ([14, 64]) For all k and k with 2 k < k, the (k 2)-th power P k 2 2k 3 induced path P 2k 3 with 2k 3 vertices is a k-leaf power but no k -leaf power. of the This means that k is the leaf rank of P k 2 2k 3. For some upper and lower bounds on the leaf 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) 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) 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) L(k). (ii) L(k) 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) L(7) and L(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

8 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 K k (which is the (k 2)-th power of the induced path P k ) 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

9 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,E T ) let T x denote the star with center x in T, i.e., T x = N T [x]. Observation 5.3 ([11]) Let G = T 2 for a tree T. Then the following conditions are equivalent: 9

10 (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 T x T y. (iv) xy is the mid-edge of a P 4 in T. Observation 5.4 Let T be a tree. Then the maximal cliques in T 2 are exactly the stars T x, x V (T), for which x is no leaf in T. Observation 5.4 is also mentioned in [42]. For the graphs G 1,...,G 5 mentioned in Theorem 5.2, see Figure 3. Figure 3: Graphs G 1 - G 8 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 (G 1,...,G 5 )-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

11 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 G 1 -G 8 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 (G 1,...,G 8 )-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 4- leaf 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 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 G 1, G 2, G 3, G 4, and G 5 in Figure 3 are not basic 4-leaf powers. Moreover, G 1 and G 4 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 G 6, G 7, and G 8 in Figure 3 are not basic 4-leaf powers. Moreover, G 6 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 K p. 11

12 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 (G 1,...,G 8 )-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 G 1,...,G 5 are responsible for the structure of the 2-connected components whereas the graphs G 6,G 7,G 8 represent the gluing conditions of the 2-connected components, i.e., blocks. Note that all forbidden subgraphs G 1 G 8 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 K 2. (ii) A plump bull is the graph resulting from the bull by replacing each of the two cutvertices by a K 2, 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 F 1,...,F 8 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

13 Figure 4: Forbidden induced subgraphs F 1,...,F 8 of 2-connected basic distance-hereditary 5-leaf powers (ii) G is (F 1,...,F 8 )-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 F 9,...,F 34 (which we do not define here - see [10] for them) reflect these conditions. A vertex v is a special vertex in graph G if N G (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 (F 1,...,F 34 )-free chordal. (iii) (a) The blocks of G are 3-leaf powers, plump darts or plump bulls. (b) For every two blocks B 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

14 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 d T (x,y) k, and, for all non-edges xy E, d T (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 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

15 Figure 5: Forbidden subgraphs H 1,...,H 6. 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 G 1,G 4,G 6 and Figure 5 for H 1 H 6 ): Theorem 6.2 The following conditions are equivalent: (i) G is a basic (6, 8)-leaf power. 15

16 (ii) G is (G 1,G 4,G 6,H 1,H 2,H 3,H 4,H 5,H 6 )-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 = (V G,E G ) is an exact k-leaf power if there is a tree T = (V T,E T ) with set V G of leaves such that xy E G if and only if d T (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. 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

17 (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 = (V G,E G ) is the k-simplicial power of graph H = (V H,E H ) if V G V H is the set of all simplicial vertices in H and for all x,y V G, xy E G if and only if d H (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, V G 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 P 3 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 = (V G,E G ) and an integer k. Question: Is there a split graph H = (V H,E H ) with V H k such that G is the 2-simplicial power of H? Theorem simplicial split graph root is NP-complete. 17

18 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

19 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 Chính Hoàng for numerous discussions and the support by Wilfrid Laurier University, Waterloo, Ontario. References [1] H.J. Bandelt, H.M. Mulder, Distance hereditary graphs, J. Combin. Theory (B), 41 (1986), [2] C. Berge, Graphs and Hypergraphs, North-Holland [3] A. Berry, A. Sigayret, Representing a concept lattice by a graph, Discrete Applied Math. 144 (2004) [4] E. Bibelnieks, P.M. Dearing, Neighborhood subtree tolerance graphs, Discrete Applied Math. 98 (2006) [5] A. Brandstädt, C. Hundt, Ptolemaic graphs and interval graphs are leaf powers, extended abstract in: Proceedings LATIN 2008, E.S. Laber et al. (Eds.), LNCS 4957, , [6] A. Brandstädt, C. Hundt, F. Mancini, P. Wagner, Rooted directed path graphs are leaf powers, manuscript 2008, submitted. [7] A. Brandstädt, V.B. Le, Structure and linear time recognition of 3-leaf powers, Information Processing Letters 98 (2006) [8] A. Brandstädt, V.B. Le, Simplicial powers of graphs, extended abstract in: Proceedings COCOA 2008, LNCS 5165, , Full version electronically available in Theor. Computer Science. [9] A. Brandstädt, V.B. Le, D. Rautenbach, Exact k-leaf powers, manuscript 2006, submitted. [10] A. Brandstädt, V.B. Le, D. Rautenbach, Distance-hereditary 5-leaf powers, Discrete Math. 309 (2009) [11] A. Brandstädt, V.B. Le, R. Sritharan, Structure and linear time recognition of 4-leaf powers, elektronically available in ACM Transactions on Algorithms, Vol. 5, No. 1, [12] A. Brandstädt, V.B. Le, J.P. Spinrad, Graph Classes: A Survey, SIAM Monographs on Discrete Math. Appl., Vol. 3, SIAM, Philadelphia (1999). [13] A. Brandstädt, P. Wagner, On (k, l)-leaf powers; extended abstract in: Proceedings of MFCS 2007, L. Kučera and A. Kučera, eds., Lecture Notes in Computer Science 4708, , Full version tentatively accepted for Discrete Applied Math. [14] A. Brandstädt, P. Wagner, On k- versus (k + 1)-leaf powers, extended abstract in: Proceedings COCOA 2008, LNCS 5165, ,

20 [15] M.W. Broin, T.J. Lowe, A dynamic programming algorithm for covering problems with (greedy) totally balanced constraint matrices, SIAM J. Alg. Disc. Meth. 7 (1986) [16] P. Buneman, A note on the metric properties of trees, J. Combin. Th. (B) 1 (1974) [17] J. Cáceres, A. Márquez, O.R. Oellermann, M.L. Puertas, Rebuilding convex sets in graphs, Discrete Math. 293 (2005), [18] M.-S. Chang, T. Ko, The 3-Steiner Root Problem; extended abstract in: Proceedings 33rd International Workshop on Graph-Theoretic Concepts in Computer Science WG 2007, LNCS 4769, , [19] M.-S. Chang, T. Ko, and H.-I. Lu, Linear time algorithms for tree root problem; extended abstract in: Proceedings SWAT 2006, LNCS 4059, , [20] Z.-Z. Chen, T. Jiang, G. Lin, Computing phylogenetic roots with bounded degrees and errors, SIAM J. Computing 32 (2003) [21] Z.-Z. Chen, T. Tsukiji, Computing bounded-degree phylogenetic roots of disconnected graphs, extended abstract in: Proceedings WG 2004, LNCS 3353, , 2004; J. Algorithms (2005), electronically available. [22] E. Dahlhaus, P. Duchet, On strongly chordal graphs, Ars Combin. 24 B (1987) [23] M. Dom, J. Guo, F. Hüffner, R. Niedermeier, Error compensation in leaf root problems, Extended abstract in: Proceedings of 15th ISAAC, LNCS 3341, , 2004; Algorithmica 44 (2006), [24] M. Dom, J. Guo, F. Hüffner, R. Niedermeier, Extending the tractability border for closest leaf powers, Extended abstract in: Proceedings of 31st Workshop on Graph-Theoretic Concepts in Computer Science WG 2005, LNCS 3787, [25] R. Fagin, Acyclic database schemes (of various degrees): A painless introduction, Proceedings CAAP 83 8th Colloquium on Trees in Algebra and Programming, LNCS 159 (1983) ed. G. Ausiello and M. Protasi, pp [26] M. Farber, Characterizations of strongly chordal graphs, Discrete Math. 43 (1983) [27] B. Farzad, L.C. Lau, V.B. Le, N.T. Nguyen, Computing Graph Roots Without Short Cycles, Proceedings of Symposium on Theoretical Aspects of Computer Science STACS 2009, Dagstuhl Seminar Proceedings 09001, pp [28] M.R. Fellows, D. Meister, F.A. Rosamond, R. Sritharan, and J.A. Telle, Leaf powers and their properties: Using the trees, Extended abstract in: Proceedings of ISAAC 2008, LNCS 5369, , [29] M. Golumbic, U. Rotics, On the clique-width of some perfect graph classes, International J. Foundat. Computer Science 11, 3 (2000) [30] F. Gurski, E. Wanke, The clique-width of tree powers and leaf power graphs, Extended abstract in: Proceedings 33rd International Workshop on Graph-Theoretic Concepts in Computer Science WG 2007, LNCS 4769, 76-85, [31] F. Harary, Graph Theory, Addison-Wesley, Massachusetts, [32] R.B. Hayward, P.E. Kearney, Investigating NeST graphs, Tech. Report TR-CS-04-93, [33] R.B. Hayward, P.E. Kearney, A. Malton, NeST graphs, Discrete Applied Math. 121 (2002) [34] E. Howorka, A characterization of distance hereditary graphs, Quart. J. Math. Oxford Ser. 2, 28 (1977) [35] E. Howorka, On metric properties of certain clique graphs, J. Combin. Th. (B) 27 (1979) [36] E. Howorka, A characterization of ptolemaic graphs, J. Graph Theory 5 (1981) [37] T. Jiang, P.E. Kearney, G.-H. Lin, Phylogenetic k-root and Steiner k-root, Extended abstract in: Proceedings of 11th Annual International Symposium on Algorithms and Computation ISAAC 2000, LNCS 1969, , [38] P.E. Kearney, D.G. Corneil, Tree powers, J. Algorithms 29 (1998) [39] W. Kennedy, Strictly chordal graphs and phylogenetic roots, Master Thesis, University of Alberta (2005). [40] W. Kennedy, G. Lin, 5-th Phylogenetic Root Construction for Strictly Chordal Graphs, Proceedings of ISAAC 2005, LNCS 3827, (2005). [41] W. Kennedy, G. Lin, G. Yan, Strictly chordal graphs are leaf powers, J. of Discrete Algorithms 4 (2006)

21 [42] L.C. Lau, Bipartite roots of graphs, In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms 2004 (SODA 2004), ACM Transactions on Algorithms, 2 (2006) [43] L.C. Lau, D.G. Corneil, Recognizing powers of proper interval, split and chordal graphs, SIAM J. Discrete Math. 18 (2004) [44] V.B. Le, N.T. Nguyen, Hardness Results and Efficient Algorithms for Graph Powers, extended abstract to appear in: Proceedings of the 35th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2009, LNCS. [45] G.-H. Lin, P.E. Kearney, T. Jiang, Phylogenetic k-root and Steiner k-root, Extended abstract in: Proceedings of ISAAC 2000, LNCS 1969, , [46] Y.-L. Lin, S.S. Skiena, Algorithms for square roots of graphs, SIAM J. Discrete Math. 8 (1995) [47] A. Lubiw, Γ-free matrices, Master Thesis, Dept. of Combinatorics and Optimization, University of Waterloo, Canada, [48] R.M. McConnell, Linear Time Recognition of Circular-Arc Graphs, Algorithmica 37 (2003) [49] R.M. McConnell, J. Spinrad, Modular decomposition and transitive orientation, Discrete Math. 201 (1999) [50] R. Motwani and M. Sudan, Computing roots of graphs is hard, Discrete Appl. Math. 54 (1994) [51] N.T. Nguyen, Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms, Ph.D. Thesis, University of Rostock, 2009, submitted. [52] N. Nishimura, P. Ragde, D.M. Thilikos, On graph powers for leaf-labeled trees, J. Algorithms 42 (2002) [53] D. Rautenbach, Some remarks about leaf roots, Discrete Math. 306, 13 (2006) [54] A. Raychaudhuri, On powers of strongly chordal and circular arc graphs, Ars Combin. 34 (1992) [55] F.S. Roberts, Indifference graphs, in F. Harary (ed.), Proof Techniques in Graph Theory, Academic Press (1969) [56] D.J. Rose, R.E. Tarjan, G.S. Lueker, Algorithmic aspects of vertex elimination on graph, SIAM J. Computing 5 (1976) [57] I.C. Ross, F. Harary, The square of a tree, Bell System Tech. J. 39 (1960) [58] J.P. Spinrad, Efficient Graph Representations, Fields Institute Monographs, Toronto [59] A. Tamir, A class of balanced matrices arising from location problems, SIAM J. Alg. Disc. Meth., 4: , [60] R.E. Tarjan, M. Yannakakis, Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Computing 13 (1984) (Addendum: SIAM J. Comput. 14 (1985) ). [61] I. Todinca, Coloring powers of graphs of bounded clique-width, In: Proceedings 29th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2003, LNCS 2880, [62] T. Tsukiji, Z.-Z. Chen, Computing phylogenetic roots with bounded degrees and errors is hard, In: Proceedings of COCOON 2004, LNCS 3106, , [63] R. Uehara, S. Toda, T. Nagoya, Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs, Discrete Aplied Math. 145, 3 (2005) [64] P. Wagner, A. Brandstädt, The Complete Inclusion Structure of Leaf Power Classes, manuscript Full version tentatively accepted for Theor. Computer Science. 21