Simplicial Powers of Graphs - Springer Link
Simplicial Powers of Graphs Andreas Brandst¨ adt and Van Bang Le Institut f¨ ur Informatik, Universit¨ at Rostock, D-18051 Rostock, Germany {ab,le}@informatik.uni-rostock.de
Abstract. In a finite simple undirected graph, a vertex is simplicial if its neighborhood is a clique. We say that, for k ≥ 2, a graph G = (VG , EG ) is the k-simplicial power of a graph H = (VH , EH ) (H a root graph of G) if VG is the set of all simplicial vertices of H, and for all distinct vertices x and y in VG , xy ∈ EG if and only if the distance in H between x and y is at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees. Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k ∈ {3, 4, 5}, k-leaf powers can be recognized in linear time, and for k ∈ {3, 4}, structural characterizations are known. For all other k, recognition and structural characterization of k-leaf powers is open. Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study ksimplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs. Keywords: Graph powers, leaf powers, simplicial powers, forbidden induced subgraph characterization, chordal graphs, block graphs, ptolemaic graphs, strongly chordal graphs.
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Introduction
Motivated by background from phylogenetic trees [3,16,35], Nishimura, Ragde and Thilikos [33] introduced the following notions: For an integer k ≥ 2, a finite undirected graph G = (VG , EG ) is a k-leaf power if there is a tree T with VG as its set of leaves such that for all distinct x, y ∈ VG , xy ∈ EG if and only if the distance between x and y in T is at most k. Then T is called a k-leaf root of G. In general, G is a leaf power if G is a k-leaf power for some k ≥ 2. Obviously, a graph is a 2-leaf power if and only if it is a disjoint union of cliques or, equivalently, it contains no induced path P3 with three vertices and two edges. In [33], a (very complicated) O(n3 ) time algorithm for recognizing 3-leaf powers and 4-leaf powers, respectively, and constructing 3-leaf roots and 4leaf roots, respectively, if they exist, was described. Recently, Chang and Ko [15] gave a linear time recognition algorithm for 5-leaf powers. Despite considerable B. Yang, D.-Z. Du, and C.A. Wang (Eds.): COCOA 2008, LNCS 5165, pp. 160–170, 2008. c Springer-Verlag Berlin Heidelberg 2008
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effort, for k ≥ 6, no characterization and no efficient recognition of k-leaf powers is known. See [6,7,9,11,12,19,34] for more information on leaf powers and in particular, for new characterizations of 3- and 4-leaf powers as well as of distancehereditary 5-leaf powers and related classes. It is known that for every k ≥ 2, k-leaf powers are strongly chordal [7] (for the definition of strongly chordal graphs see section 2). In [4], Bibelnieks and Dearing introduced and studied so-called NeST graphs (i.e., neighborhood subtree tolerance graphs); for constant tolerances these are exactly the induced subgraphs of powers of trees [8,23] which are closely related to k-leaf powers (see Proposition 1). In [4], an example of a graph is given which is strongly chordal but no fixed tolerance NeST graph (i.e., no k-leaf power for any k), and in [23] this is slightly generalized; [23] mentions the open problem of characterizing fixed tolerance NeST graphs. Definition 1 gives the key notion of this paper, namely k-simplicial powers of graphs which generalizes the notion of k-leaf powers of trees in a very natural way and which is also of independent interest. A vertex is simplicial if its neighborhood is a clique. Simplicial vertices of degree one are called leaves. Definition 1. For any integer k ≥ 1, graph G = (VG , EG ) is the k-simplicial power of graph H = (VH , EH ) if VG ⊆ VH is the set of all simplicial vertices in H and for all distinct vertices x, y ∈ VG , xy ∈ EG if and only if the distance in H between x and y is at most k. Such a graph H is a k-simplicial root of G. If G is the k-simplicial power of H and if, in addition, VG 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. Since trees and block graphs (i.e., those graphs whose 2-connected components are cliques) have very similar metric properties (see Theorem 2), it is natural to study k-simplicial powers of block graphs. In particular, the main motivation of this paper comes from Theorem 6 which claims that for any k ≥ 2, a graph is the k-leaf power of a tree if and only if it is the (k − 1)-simplicial power of a claw-free block graph. Thus, our focus is on simplicial powers of block graphs but we also consider simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs. Due to space limitations in this extended abstract, proofs are omitted.
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Basic Notions and Results
Throughout this paper, let G = (VG , EG ) denote a finite undirected graph without loops and multiple edges, with vertex set VG and edge set EG . Moreover, we assume connectedness unless stated otherwise. For a vertex v ∈ VG , let NG (v) = {u | uv ∈ EG } denote the neighborhood of v in G, and let NG [v] = {v} ∪ NG (v) denote the closed neighborhood of v in G. The degree degG (v) of a vertex v is the number of its neighbors, i.e., degG (v) = |NG (v)|. The complement graph of G is denoted by G. A clique is a set of mutually adjacent vertices. A stable set is a set of mutually non-adjacent vertices.
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A cut vertex is a vertex whose removal increases the number of connected components. A connected graph is 2-connected if it has no cut vertex. As usual, the maximal induced 2-connected subgraphs of G are the blocks (or 2-connected components) of G. A block of G which contains at most one cut vertex is an endblock. For U ⊆ V , let G[U ] denote the subgraph of G induced by U . For a set F of graphs, a graph is F -free if none of its induced subgraphs is in F . Two vertices x, y ∈ V are true twins if NG [x] = NG [y]. A vertex set U ⊆ VG is a module of G if U ⊆ NG (v) or U ∩ NG (v) = ∅ for all v ∈ VG \ U . A homogeneous set of G is a module which consists of at least two, but not all vertices of G. A clique module in G is a module which is a clique in G. Obviously, true twins form a clique module. Replacing a vertex v in a graph G by a graph H (or substituting H into v) results in the graph obtained from G[VG \ {v}] ∪ H by adding all edges between vertices in NG (v) and vertices in VH . For a positive integer k ≥ 1, let Pk denote the chordless path with k vertices and k − 1 edges, and for k ≥ 3, let Ck denote the 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 Kr,s ; the K1,3 is also called the claw. For k ≥ 3, let Sk denote the (complete) sun with 2k vertices u1 , . . . , uk and w1 , . . . , wk such that u1 , . . . , uk is a clique, w1 , . . . , wk is a stable set and for i ∈ {1, . . . , k}, wi is adjacent to exactly ui and ui+1 (index arithmetic modulo k). A graph is sun-free if it contains no induced Sk for any k ≥ 3. A graph is chordal if it contains no induced Ck for any k ≥ 4. A graph is strongly chordal if it is chordal and sun-free. It is known that leaf powers are strongly chordal (cf. [7], Proposition 3). A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. It is well known that G is a split graph if and only if G and its complement graph G are chordal. A graph is ptolemaic if it is chordal and gem-free (see Figure 1 for the gem). A connected graph is a block graph if each of its blocks is a clique. Clearly, block graphs are ptolemaic but not vice versa. As block graphs will play a crucial role in this paper, we give here some well-known characterizations of them; the equivalence (i) ⇔ (ii) in Theorem 1 is Theorem 3.5 in [24], and the equivalence (i) ⇔ (iii) can be easily seen, e.g., by [11, Observation 3]. Theorem 1. For every graph G, the following statements are equivalent: (i) G is a block graph. (ii) G is the intersection graph of the blocks of some graph. (iii) G is chordal and diamond-free. Let dG (x, y) denote the distance in G between x and y (i.e., the minimum number of edges of a path in G connecting x and y). A graph G is distance hereditary if in every connected induced subgraph H of G, the distance function is the same as in G, i.e., dH = dG |VH . In [25] it was shown that a chordal graph is distance hereditary if and only if it is gem-free. In particular, distance-hereditary and chordal graphs, i.e., ptolemaic graphs, are strongly chordal but not vice versa. k k ) with xy ∈ EG if and only if dG (x, y) ≤ k denote the k-th Let Gk = (VG , EG power of G. See [1,2,5,10,18,20,21,31,32] for basic properties of powers of strongly chordal graphs (chordal graphs, distance-hereditary graphs, respectively).
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diamond
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3-sun
Fig. 1. The bull, diamond, dart, gem, and 3-sun
Buneman’s four-point condition (∗) for distances in graphs requires that for every four vertices u, v, x, y, the following inequality holds: (∗) dG (u, v) + dG (x, y) ≤ max{dG (u, x) + dG (v, y), dG (u, y) + dG (v, x)}. The following well-known results show that trees and block graphs have very similar metric properties. Theorem 2. Let G be a connected graph. (i) [13] G is a tree if and only if G is triangle-free and fulfills the four-point condition (∗). (ii) [26] G is a block graph if and only if G satisfies (∗). Finally, we mention some fundamental but simple properties, among them the following result characterizing 3-leaf powers: Theorem 3 ([7,19,34]). For every graph G, the following 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. In [30], the following notion for k ≥ 1 is defined: A tree T = (VT , ET ) is a k-th Steiner root of the graph G = (VG , EG ) if VG ⊆ VT and xy ∈ EG if and only if dT (x, y) ≤ k. In this case, G is a k-th Steiner power. In [11], we say that a graph G is a basic k-leaf power if G has a k-leaf root T such that no two leaves of T are attached to the same parent vertex in T (a so-called basic k-leaf root). Obviously, for k ≥ 2, the set of leaves having the same parent node in T form a clique, and G is a k-leaf power if and only if G results from a basic k-leaf power by substituting cliques into its vertices. If T is a basic k-leaf root of G then T minus its leaves is a (k − 2)-th Steiner root of G. Summarising, the following obvious equivalences hold: Proposition 1. For a graph G, the following are equivalent for all k ≥ 2 : (i) G has a k-th Steiner root. (ii) G is an induced subgraph of the k-th power of a tree. (iii) G is a basic (k + 2)-leaf power.
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Simplicial Powers Versus Leaf Powers
Recall that the notion of k-simplicial powers (see Definition 1) is the key notion of this paper. 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 is P3 -free. As Proposition 2 shows, every graph is the 2-simplicial power of some split graph. Thus, the notion of k-simplicial power is only interesting for some very restricted classes of root graphs. Proposition 2. Every graph is (i) the 2-simplicial power of a split graph, and (ii) the 4-leaf power of a bipartite graph. Since in the proof of Proposition 2 (i), for given graph G a split graph G is constructed which might be exponentially larger than G, Proposition 2 (i) suggests the following problem: 2-simplicial split graph root Instance: A graph G = (VG , EG ) and an integer N . Question: Does there exist a split graph H = (VH , EH ) with |VH | ≤ N such that G is the 2-simplicial power of H? By reducing the problem intersection graph basis ([22, GT59]) to our problem, we obtain: Theorem 4. 2-simplicial split graph root is NP-complete. Let G = (VG , EG ) be a graph. Its line graph L(G) has EG as its vertices, and two edges e, e are adjacent in L(G) if and only if e ∩ e = ∅. Theorem 5 ([24], Theorem 8.5). A graph is the line graph of a tree if and only if it is a claw-free block graph. The subsequent Theorem 6 was the main motivation for this paper. Theorem 6. For k ≥ 2, a graph is the k-leaf power of a tree if and only if it is the (k − 1)-simplicial power of a claw-free block graph. Corollary 1. The class of k-simplicial powers of block graphs contains all t-leaf powers for t ≤ k + 1.
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2-Simplicial Powers of Some Subclasses of Chordal Graphs
By Theorem 6, every 3-leaf power is the 2-simplicial power of a claw-free block graph. Theorem 7 characterizes the larger class of 2-simplicial powers of block graphs as the (dart,gem)-free chordal graphs. Note that this graph class appears in other contexts as well:
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– In [14], in connection with convexity of graphs, the notion of contour vertices is defined, and it is shown that a connected graph G has the property that for all convex sets S in G, the contour vertices of S coincide with the eccentric vertices of S if and only if G is (dart, gem)-free chordal. – In [28], so-called strictly chordal graphs are introduced via rather complicated hypergraph properties, and it is shown that these graphs are leaf powers. It turns out that a graph is strictly chordal if and only if it is (dart,gem)-free chordal [27]. – In [12], the notion of k-leaf root and k-leaf power is modified in the following way: For k ≥ 2 and > k, a tree T is a (k, )-leaf root of a graph G = (VG , EG ) if VG is the set of leaves of T , for all edges xy ∈ EG , dT (x, y) ≤ k and, for all non-edges xy ∈ EG , dT (x, y) ≥ . A graph G is a (k, )-leaf power if it has a (k, )-leaf root. Thus, every k-leaf power is a (k, k + 1)-leaf power. Then, it is shown in [12]: Every block graph is a (4, 6)-leaf power, and a (4, 6)-leaf root of it can be determined in linear time. Moreover, G is a (4, 6)-leaf power if and only if G is (dart,gem)-free chordal. Recall that Theorem 3 characterizes 3-leaf powers (of trees) as the (bull,dart,gem)free chordal graphs. Comparing Theorem 7 with Theorem 3 shows how natural the concept of simplicial powers of block graphs fits within the world of leaf powers. Theorem 7. 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 results from substituting cliques into the vertices of a block graph. (iv) G is a (4, 6)-leaf power. Theorem 8. For every graph G, the following statements are equivalent: (i) G is the 2-simplicial power of a ptolemaic graph. (ii) G is the 2-simplicial power of a ptolemaic split graph. (iii) G is ptolemaic. An analogous equivalence holds if in Theorem 8, “ptolemaic” is replaced by “strongly chordal” in all three statements.
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Simplicial Powers of Block Graphs
Theorem 6 indicates the close relationship between leaf powers (of trees) and simplicial powers of (claw-free) block graphs. However, the larger class of simplicial powers of (not necessarily claw-free) block graphs is of independent interest. As already mentioned (see [11] and Proposition 1), leaf powers of trees are exactly those graphs obtainable from an induced subgraph of a tree power by replacing vertices by cliques. A similar statement is true for simplicial powers of block graphs; it is based on the following notion. Definition 2. A graph G is a basic k-simplicial power of a block graph if G admits a k-simplicial block graph root R in which each block contains at most one simplicial vertex.
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Examples of basic k-simplicial powers of block graphs include block graphs and k-leaf powers. Obviously, every simplicial power of a block graph is obtained from a basic simplicial power of a block graph by replacing vertices by cliques. Moreover, if G = (VG , EG ) is a basic k-simplicial power, then any (connected) induced subgraph of G is also a basic k-simplicial power of a block graph: If R = (VR , ER ) is a basic k-simplicial block graph root of G and G is a subgraph of G induced by S ⊆ VG , then it can be easily seen that the smallest connected subgraph of R containing S is a basic k-simplicial block graph root of G . Theorem 9. Let k ≥ 2 be an integer. A graph is a basic k-simplicial power of a block graph if and only if it is an induced subgraph of the (k − 1)-th power of a block graph. The proof of Theorem 9 shows directly: Corollary 2. Let k ≥ 2 be an integer. A basic k-simplicial power of a block graph is the (k − 1)-th power of a block graph if and only if it admits a basic k-simplicial block graph root in which each block contains exactly one simplicial vertex. In the rest of this section we will describe the basic 3-simplicial powers of block graphs in more detail. Definition 3. A maximal clique Q in a graph G = (VG , EG ) is special if for all x, y ∈ VG − Q having a common neighbor in Q, N (x) ∩ Q = N (y) ∩ Q or |N (x) ∩ Q| = 1 or |N (y) ∩ Q| = 1. A vertex v of G is special if N [v] is a special clique in G. Note that a special vertex is in particular simplicial. It turns out that special vertices play an important role in recognizing 2-connected basic 3-simplicial powers of block graphs. For a description of 2-connected basic 3-simplicial powers of block graphs, we need the following notion. A split of a graph G = (VG , EG ) is a partition into two disjoint sets V1 and V2 such that |V1 | ≥ 2, |V2 | ≥ 2 and the set of edges of G between V1 and V2 forms a complete bipartite graph. Graphs without split are called prime. A simple split decomposition of G by the split (V1 , V2 ) is the decomposition of G into two graph G1 and G2 where Gi is obtained from the subgraph of G induced by Vi and an additional vertex (a so-called marker) v by adding all edges between v and those vertices in Vi which have a neighbor in G − Vi . Split decomposition can be computed in linear time [17]. We characterize 2-connected basic 3-simplicial powers of block graphs by reducing to smaller ones as follows. Theorem 10. A 2-connected graph G = (VG , EG ) is a basic 3-simplicial power of a block graph if and only if
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(i) G is the square of a block graph, or (ii) G has a special vertex v such that NG (v) = NG (x) ∩ NG (y) for some nonadjacent vertices x and y, and G − v is a 2-connected basic 3-simplicial power of a block graph, or (iii) G admits a split (V1 , V2 ) such that G1 and G2 are 2-connected basic 3simplicial powers of block graphs and the marked vertex is special in both G1 and G2 . Theorem 10 gives a recursive procedure that checks in time O(n3 ) whether a 2-connected chordal graph G with n vertices is a basic 3-simplicial power of a block graph: Checking whether G is the square of a block graph can be done in linear time by a result in [29]. If G is not the square of a block graph then check whether G satisfies (ii) or (iii). If yes, recursively check the corresponding 2-connected graphs G − v, and G1 and G2 , respectively. Whether a maximal clique is special can be easily checked in time O(n2 ), the at most n maximal cliques in a chordal graph can be found in linear time, and checking (ii) and (iii) can be done in time O(n3 ). Observation 1 Every 2-connected basic 3-simplicial power of a block graph G = (VG , EG ) admits a basic 3-simplicial block graph root R such that, for all special vertices c of G and all x ∈ VG − c, dR (c, x) ≥ 3. Theorem 11. For every graph G, the following statements are equivalent: (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. Corollary 3. 3-simplicial powers of block graphs can be recognized efficiently.
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Fig. 2. Forbidden subgraphs G1 , . . . , G9 characterize induced subgraphs of squares of block graphs
In the full version of [12], induced subgraphs of squares of block graphs (see Theorem 11 (ii)) are also characterized in terms of forbidden subgraphs (see Figure 2), and similarly as for k = 2 in Theorem 7, 3-simplicial powers of a block graph are closely related to (6,8)-leaf powers as described in Theorems 11 and 12.
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Theorem 12 ([12]). For every graph G, the following are equivalent: (i) G is a basic (6, 8)-leaf power. (ii) G is an induced subgraph of the square of a block graph. (iii) G is (G1 , G2 , . . . , G9 )-free chordal. This characterization is inspired by the corresponding results for 4-leaf powers in [11,34]. The graphs G1 , G2 , G4 , G5 , G6 , G7 express separator properties of induced subgraphs of squares of block graphs which are 2-connected, and the graphs G3 , G8 , G9 express the gluing conditions for the 2-connected components of such graphs.
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Conclusion
Simplicial powers of block graphs (ptolemaic graphs, strongly chordal graphs, respectively) are a natural generalization of leaf powers. There are close connections between k-leaf powers, (k, k + 2)-leaf powers and simplicial powers of block graphs such as described in Theorems 6, 7, 11 and 12. While every graph is the 2-simplicial power of a split graph and the 4-leaf power of a bipartite graph, 2-simplicial powers of ptolemaic graphs (strongly chordal graphs, respectively) are ptolemaic (strongly chordal, respectively). Since leaf powers are strongly chordal (but not vice versa), our results on simplicial powers of block graphs and of ptolemaic graphs might shed new light on the open problem of characterizing k-leaf powers for k ≥ 5 and of characterizing leaf powers in general. We gave various characterizations of classes defined as simplicial powers of certain graph classes. In particular, we obtained the following hierarchy: - 3-leaf powers (which are exactly the (bull,dart,gem)-free chordal graphs) are a proper subclass of - 2-simplicial powers of block graphs (which are exactly the (dart,gem)-free chordal graphs), and these are in turn a proper subclass of - 2-simplicial powers of ptolemaic graphs (which are exactly the gem-free chordal graphs). The class of 3-simplicial powers of block graphs is an interesting generalization of 4-leaf powers and is characterized in Theorems 11 and 12. We hope that our approach will lead to new insights about the structure of k-leaf powers for k ≥ 5.
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Abstract. In a finite simple undirected graph, a vertex is simplicial if its neighborhood is a clique. We say that, for k ≥ 2, a graph G = (VG , EG ) is the k-simplicial power of a graph H = (VH , EH ) (H a root graph of G) if VG is the set of all simplicial vertices of H, and for all distinct vertices x and y in VG , xy ∈ EG if and only if the distance in H between x and y is at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees. Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k ∈ {3, 4, 5}, k-leaf powers can be recognized in linear time, and for k ∈ {3, 4}, structural characterizations are known. For all other k, recognition and structural characterization of k-leaf powers is open. Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study ksimplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs. Keywords: Graph powers, leaf powers, simplicial powers, forbidden induced subgraph characterization, chordal graphs, block graphs, ptolemaic graphs, strongly chordal graphs.
1
Introduction
Motivated by background from phylogenetic trees [3,16,35], Nishimura, Ragde and Thilikos [33] introduced the following notions: For an integer k ≥ 2, a finite undirected graph G = (VG , EG ) is a k-leaf power if there is a tree T with VG as its set of leaves such that for all distinct x, y ∈ VG , xy ∈ EG if and only if the distance between x and y in T is at most k. Then T is called a k-leaf root of G. In general, G is a leaf power if G is a k-leaf power for some k ≥ 2. Obviously, a graph is a 2-leaf power if and only if it is a disjoint union of cliques or, equivalently, it contains no induced path P3 with three vertices and two edges. In [33], a (very complicated) O(n3 ) time algorithm for recognizing 3-leaf powers and 4-leaf powers, respectively, and constructing 3-leaf roots and 4leaf roots, respectively, if they exist, was described. Recently, Chang and Ko [15] gave a linear time recognition algorithm for 5-leaf powers. Despite considerable B. Yang, D.-Z. Du, and C.A. Wang (Eds.): COCOA 2008, LNCS 5165, pp. 160–170, 2008. c Springer-Verlag Berlin Heidelberg 2008
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effort, for k ≥ 6, no characterization and no efficient recognition of k-leaf powers is known. See [6,7,9,11,12,19,34] for more information on leaf powers and in particular, for new characterizations of 3- and 4-leaf powers as well as of distancehereditary 5-leaf powers and related classes. It is known that for every k ≥ 2, k-leaf powers are strongly chordal [7] (for the definition of strongly chordal graphs see section 2). In [4], Bibelnieks and Dearing introduced and studied so-called NeST graphs (i.e., neighborhood subtree tolerance graphs); for constant tolerances these are exactly the induced subgraphs of powers of trees [8,23] which are closely related to k-leaf powers (see Proposition 1). In [4], an example of a graph is given which is strongly chordal but no fixed tolerance NeST graph (i.e., no k-leaf power for any k), and in [23] this is slightly generalized; [23] mentions the open problem of characterizing fixed tolerance NeST graphs. Definition 1 gives the key notion of this paper, namely k-simplicial powers of graphs which generalizes the notion of k-leaf powers of trees in a very natural way and which is also of independent interest. A vertex is simplicial if its neighborhood is a clique. Simplicial vertices of degree one are called leaves. Definition 1. For any integer k ≥ 1, graph G = (VG , EG ) is the k-simplicial power of graph H = (VH , EH ) if VG ⊆ VH is the set of all simplicial vertices in H and for all distinct vertices x, y ∈ VG , xy ∈ EG if and only if the distance in H between x and y is at most k. Such a graph H is a k-simplicial root of G. If G is the k-simplicial power of H and if, in addition, VG 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. Since trees and block graphs (i.e., those graphs whose 2-connected components are cliques) have very similar metric properties (see Theorem 2), it is natural to study k-simplicial powers of block graphs. In particular, the main motivation of this paper comes from Theorem 6 which claims that for any k ≥ 2, a graph is the k-leaf power of a tree if and only if it is the (k − 1)-simplicial power of a claw-free block graph. Thus, our focus is on simplicial powers of block graphs but we also consider simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs. Due to space limitations in this extended abstract, proofs are omitted.
2
Basic Notions and Results
Throughout this paper, let G = (VG , EG ) denote a finite undirected graph without loops and multiple edges, with vertex set VG and edge set EG . Moreover, we assume connectedness unless stated otherwise. For a vertex v ∈ VG , let NG (v) = {u | uv ∈ EG } denote the neighborhood of v in G, and let NG [v] = {v} ∪ NG (v) denote the closed neighborhood of v in G. The degree degG (v) of a vertex v is the number of its neighbors, i.e., degG (v) = |NG (v)|. The complement graph of G is denoted by G. A clique is a set of mutually adjacent vertices. A stable set is a set of mutually non-adjacent vertices.
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A cut vertex is a vertex whose removal increases the number of connected components. A connected graph is 2-connected if it has no cut vertex. As usual, the maximal induced 2-connected subgraphs of G are the blocks (or 2-connected components) of G. A block of G which contains at most one cut vertex is an endblock. For U ⊆ V , let G[U ] denote the subgraph of G induced by U . For a set F of graphs, a graph is F -free if none of its induced subgraphs is in F . Two vertices x, y ∈ V are true twins if NG [x] = NG [y]. A vertex set U ⊆ VG is a module of G if U ⊆ NG (v) or U ∩ NG (v) = ∅ for all v ∈ VG \ U . A homogeneous set of G is a module which consists of at least two, but not all vertices of G. A clique module in G is a module which is a clique in G. Obviously, true twins form a clique module. Replacing a vertex v in a graph G by a graph H (or substituting H into v) results in the graph obtained from G[VG \ {v}] ∪ H by adding all edges between vertices in NG (v) and vertices in VH . For a positive integer k ≥ 1, let Pk denote the chordless path with k vertices and k − 1 edges, and for k ≥ 3, let Ck denote the 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 Kr,s ; the K1,3 is also called the claw. For k ≥ 3, let Sk denote the (complete) sun with 2k vertices u1 , . . . , uk and w1 , . . . , wk such that u1 , . . . , uk is a clique, w1 , . . . , wk is a stable set and for i ∈ {1, . . . , k}, wi is adjacent to exactly ui and ui+1 (index arithmetic modulo k). A graph is sun-free if it contains no induced Sk for any k ≥ 3. A graph is chordal if it contains no induced Ck for any k ≥ 4. A graph is strongly chordal if it is chordal and sun-free. It is known that leaf powers are strongly chordal (cf. [7], Proposition 3). A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. It is well known that G is a split graph if and only if G and its complement graph G are chordal. A graph is ptolemaic if it is chordal and gem-free (see Figure 1 for the gem). A connected graph is a block graph if each of its blocks is a clique. Clearly, block graphs are ptolemaic but not vice versa. As block graphs will play a crucial role in this paper, we give here some well-known characterizations of them; the equivalence (i) ⇔ (ii) in Theorem 1 is Theorem 3.5 in [24], and the equivalence (i) ⇔ (iii) can be easily seen, e.g., by [11, Observation 3]. Theorem 1. For every graph G, the following statements are equivalent: (i) G is a block graph. (ii) G is the intersection graph of the blocks of some graph. (iii) G is chordal and diamond-free. Let dG (x, y) denote the distance in G between x and y (i.e., the minimum number of edges of a path in G connecting x and y). A graph G is distance hereditary if in every connected induced subgraph H of G, the distance function is the same as in G, i.e., dH = dG |VH . In [25] it was shown that a chordal graph is distance hereditary if and only if it is gem-free. In particular, distance-hereditary and chordal graphs, i.e., ptolemaic graphs, are strongly chordal but not vice versa. k k ) with xy ∈ EG if and only if dG (x, y) ≤ k denote the k-th Let Gk = (VG , EG power of G. See [1,2,5,10,18,20,21,31,32] for basic properties of powers of strongly chordal graphs (chordal graphs, distance-hereditary graphs, respectively).
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diamond
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3-sun
Fig. 1. The bull, diamond, dart, gem, and 3-sun
Buneman’s four-point condition (∗) for distances in graphs requires that for every four vertices u, v, x, y, the following inequality holds: (∗) dG (u, v) + dG (x, y) ≤ max{dG (u, x) + dG (v, y), dG (u, y) + dG (v, x)}. The following well-known results show that trees and block graphs have very similar metric properties. Theorem 2. Let G be a connected graph. (i) [13] G is a tree if and only if G is triangle-free and fulfills the four-point condition (∗). (ii) [26] G is a block graph if and only if G satisfies (∗). Finally, we mention some fundamental but simple properties, among them the following result characterizing 3-leaf powers: Theorem 3 ([7,19,34]). For every graph G, the following 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. In [30], the following notion for k ≥ 1 is defined: A tree T = (VT , ET ) is a k-th Steiner root of the graph G = (VG , EG ) if VG ⊆ VT and xy ∈ EG if and only if dT (x, y) ≤ k. In this case, G is a k-th Steiner power. In [11], we say that a graph G is a basic k-leaf power if G has a k-leaf root T such that no two leaves of T are attached to the same parent vertex in T (a so-called basic k-leaf root). Obviously, for k ≥ 2, the set of leaves having the same parent node in T form a clique, and G is a k-leaf power if and only if G results from a basic k-leaf power by substituting cliques into its vertices. If T is a basic k-leaf root of G then T minus its leaves is a (k − 2)-th Steiner root of G. Summarising, the following obvious equivalences hold: Proposition 1. For a graph G, the following are equivalent for all k ≥ 2 : (i) G has a k-th Steiner root. (ii) G is an induced subgraph of the k-th power of a tree. (iii) G is a basic (k + 2)-leaf power.
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Simplicial Powers Versus Leaf Powers
Recall that the notion of k-simplicial powers (see Definition 1) is the key notion of this paper. 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 is P3 -free. As Proposition 2 shows, every graph is the 2-simplicial power of some split graph. Thus, the notion of k-simplicial power is only interesting for some very restricted classes of root graphs. Proposition 2. Every graph is (i) the 2-simplicial power of a split graph, and (ii) the 4-leaf power of a bipartite graph. Since in the proof of Proposition 2 (i), for given graph G a split graph G is constructed which might be exponentially larger than G, Proposition 2 (i) suggests the following problem: 2-simplicial split graph root Instance: A graph G = (VG , EG ) and an integer N . Question: Does there exist a split graph H = (VH , EH ) with |VH | ≤ N such that G is the 2-simplicial power of H? By reducing the problem intersection graph basis ([22, GT59]) to our problem, we obtain: Theorem 4. 2-simplicial split graph root is NP-complete. Let G = (VG , EG ) be a graph. Its line graph L(G) has EG as its vertices, and two edges e, e are adjacent in L(G) if and only if e ∩ e = ∅. Theorem 5 ([24], Theorem 8.5). A graph is the line graph of a tree if and only if it is a claw-free block graph. The subsequent Theorem 6 was the main motivation for this paper. Theorem 6. For k ≥ 2, a graph is the k-leaf power of a tree if and only if it is the (k − 1)-simplicial power of a claw-free block graph. Corollary 1. The class of k-simplicial powers of block graphs contains all t-leaf powers for t ≤ k + 1.
4
2-Simplicial Powers of Some Subclasses of Chordal Graphs
By Theorem 6, every 3-leaf power is the 2-simplicial power of a claw-free block graph. Theorem 7 characterizes the larger class of 2-simplicial powers of block graphs as the (dart,gem)-free chordal graphs. Note that this graph class appears in other contexts as well:
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– In [14], in connection with convexity of graphs, the notion of contour vertices is defined, and it is shown that a connected graph G has the property that for all convex sets S in G, the contour vertices of S coincide with the eccentric vertices of S if and only if G is (dart, gem)-free chordal. – In [28], so-called strictly chordal graphs are introduced via rather complicated hypergraph properties, and it is shown that these graphs are leaf powers. It turns out that a graph is strictly chordal if and only if it is (dart,gem)-free chordal [27]. – In [12], the notion of k-leaf root and k-leaf power is modified in the following way: For k ≥ 2 and > k, a tree T is a (k, )-leaf root of a graph G = (VG , EG ) if VG is the set of leaves of T , for all edges xy ∈ EG , dT (x, y) ≤ k and, for all non-edges xy ∈ EG , dT (x, y) ≥ . A graph G is a (k, )-leaf power if it has a (k, )-leaf root. Thus, every k-leaf power is a (k, k + 1)-leaf power. Then, it is shown in [12]: Every block graph is a (4, 6)-leaf power, and a (4, 6)-leaf root of it can be determined in linear time. Moreover, G is a (4, 6)-leaf power if and only if G is (dart,gem)-free chordal. Recall that Theorem 3 characterizes 3-leaf powers (of trees) as the (bull,dart,gem)free chordal graphs. Comparing Theorem 7 with Theorem 3 shows how natural the concept of simplicial powers of block graphs fits within the world of leaf powers. Theorem 7. 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 results from substituting cliques into the vertices of a block graph. (iv) G is a (4, 6)-leaf power. Theorem 8. For every graph G, the following statements are equivalent: (i) G is the 2-simplicial power of a ptolemaic graph. (ii) G is the 2-simplicial power of a ptolemaic split graph. (iii) G is ptolemaic. An analogous equivalence holds if in Theorem 8, “ptolemaic” is replaced by “strongly chordal” in all three statements.
5
Simplicial Powers of Block Graphs
Theorem 6 indicates the close relationship between leaf powers (of trees) and simplicial powers of (claw-free) block graphs. However, the larger class of simplicial powers of (not necessarily claw-free) block graphs is of independent interest. As already mentioned (see [11] and Proposition 1), leaf powers of trees are exactly those graphs obtainable from an induced subgraph of a tree power by replacing vertices by cliques. A similar statement is true for simplicial powers of block graphs; it is based on the following notion. Definition 2. A graph G is a basic k-simplicial power of a block graph if G admits a k-simplicial block graph root R in which each block contains at most one simplicial vertex.
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Examples of basic k-simplicial powers of block graphs include block graphs and k-leaf powers. Obviously, every simplicial power of a block graph is obtained from a basic simplicial power of a block graph by replacing vertices by cliques. Moreover, if G = (VG , EG ) is a basic k-simplicial power, then any (connected) induced subgraph of G is also a basic k-simplicial power of a block graph: If R = (VR , ER ) is a basic k-simplicial block graph root of G and G is a subgraph of G induced by S ⊆ VG , then it can be easily seen that the smallest connected subgraph of R containing S is a basic k-simplicial block graph root of G . Theorem 9. Let k ≥ 2 be an integer. A graph is a basic k-simplicial power of a block graph if and only if it is an induced subgraph of the (k − 1)-th power of a block graph. The proof of Theorem 9 shows directly: Corollary 2. Let k ≥ 2 be an integer. A basic k-simplicial power of a block graph is the (k − 1)-th power of a block graph if and only if it admits a basic k-simplicial block graph root in which each block contains exactly one simplicial vertex. In the rest of this section we will describe the basic 3-simplicial powers of block graphs in more detail. Definition 3. A maximal clique Q in a graph G = (VG , EG ) is special if for all x, y ∈ VG − Q having a common neighbor in Q, N (x) ∩ Q = N (y) ∩ Q or |N (x) ∩ Q| = 1 or |N (y) ∩ Q| = 1. A vertex v of G is special if N [v] is a special clique in G. Note that a special vertex is in particular simplicial. It turns out that special vertices play an important role in recognizing 2-connected basic 3-simplicial powers of block graphs. For a description of 2-connected basic 3-simplicial powers of block graphs, we need the following notion. A split of a graph G = (VG , EG ) is a partition into two disjoint sets V1 and V2 such that |V1 | ≥ 2, |V2 | ≥ 2 and the set of edges of G between V1 and V2 forms a complete bipartite graph. Graphs without split are called prime. A simple split decomposition of G by the split (V1 , V2 ) is the decomposition of G into two graph G1 and G2 where Gi is obtained from the subgraph of G induced by Vi and an additional vertex (a so-called marker) v by adding all edges between v and those vertices in Vi which have a neighbor in G − Vi . Split decomposition can be computed in linear time [17]. We characterize 2-connected basic 3-simplicial powers of block graphs by reducing to smaller ones as follows. Theorem 10. A 2-connected graph G = (VG , EG ) is a basic 3-simplicial power of a block graph if and only if
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(i) G is the square of a block graph, or (ii) G has a special vertex v such that NG (v) = NG (x) ∩ NG (y) for some nonadjacent vertices x and y, and G − v is a 2-connected basic 3-simplicial power of a block graph, or (iii) G admits a split (V1 , V2 ) such that G1 and G2 are 2-connected basic 3simplicial powers of block graphs and the marked vertex is special in both G1 and G2 . Theorem 10 gives a recursive procedure that checks in time O(n3 ) whether a 2-connected chordal graph G with n vertices is a basic 3-simplicial power of a block graph: Checking whether G is the square of a block graph can be done in linear time by a result in [29]. If G is not the square of a block graph then check whether G satisfies (ii) or (iii). If yes, recursively check the corresponding 2-connected graphs G − v, and G1 and G2 , respectively. Whether a maximal clique is special can be easily checked in time O(n2 ), the at most n maximal cliques in a chordal graph can be found in linear time, and checking (ii) and (iii) can be done in time O(n3 ). Observation 1 Every 2-connected basic 3-simplicial power of a block graph G = (VG , EG ) admits a basic 3-simplicial block graph root R such that, for all special vertices c of G and all x ∈ VG − c, dR (c, x) ≥ 3. Theorem 11. For every graph G, the following statements are equivalent: (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. Corollary 3. 3-simplicial powers of block graphs can be recognized efficiently.
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Fig. 2. Forbidden subgraphs G1 , . . . , G9 characterize induced subgraphs of squares of block graphs
In the full version of [12], induced subgraphs of squares of block graphs (see Theorem 11 (ii)) are also characterized in terms of forbidden subgraphs (see Figure 2), and similarly as for k = 2 in Theorem 7, 3-simplicial powers of a block graph are closely related to (6,8)-leaf powers as described in Theorems 11 and 12.
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Theorem 12 ([12]). For every graph G, the following are equivalent: (i) G is a basic (6, 8)-leaf power. (ii) G is an induced subgraph of the square of a block graph. (iii) G is (G1 , G2 , . . . , G9 )-free chordal. This characterization is inspired by the corresponding results for 4-leaf powers in [11,34]. The graphs G1 , G2 , G4 , G5 , G6 , G7 express separator properties of induced subgraphs of squares of block graphs which are 2-connected, and the graphs G3 , G8 , G9 express the gluing conditions for the 2-connected components of such graphs.
6
Conclusion
Simplicial powers of block graphs (ptolemaic graphs, strongly chordal graphs, respectively) are a natural generalization of leaf powers. There are close connections between k-leaf powers, (k, k + 2)-leaf powers and simplicial powers of block graphs such as described in Theorems 6, 7, 11 and 12. While every graph is the 2-simplicial power of a split graph and the 4-leaf power of a bipartite graph, 2-simplicial powers of ptolemaic graphs (strongly chordal graphs, respectively) are ptolemaic (strongly chordal, respectively). Since leaf powers are strongly chordal (but not vice versa), our results on simplicial powers of block graphs and of ptolemaic graphs might shed new light on the open problem of characterizing k-leaf powers for k ≥ 5 and of characterizing leaf powers in general. We gave various characterizations of classes defined as simplicial powers of certain graph classes. In particular, we obtained the following hierarchy: - 3-leaf powers (which are exactly the (bull,dart,gem)-free chordal graphs) are a proper subclass of - 2-simplicial powers of block graphs (which are exactly the (dart,gem)-free chordal graphs), and these are in turn a proper subclass of - 2-simplicial powers of ptolemaic graphs (which are exactly the gem-free chordal graphs). The class of 3-simplicial powers of block graphs is an interesting generalization of 4-leaf powers and is characterized in Theorems 11 and 12. We hope that our approach will lead to new insights about the structure of k-leaf powers for k ≥ 5.
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