Introducing Subclasses of Basic Chordal Graphs

Introducing subclasses of basic chordal graphs

Pablo De Caria 1 Marisa Gutierrez 2

CONICET/ Universidad Nacional de La Plata, Argentina

Abstract Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were deﬁned as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we ﬁnd some other properties they have, mostly involving clique graphs. Keywords: Chordal graph, dually chordal graph, clique tree, compatible tree.

1 Introduction

1.1 Deﬁnitions For a graph G, V (G) is the set of its vertices and E(G) is the set of its edges. A clique is a maximal set of pairwise adjacent vertices. The family of cliques of G is denoted by C(G). For v ∈ V (G), Cv will denote the family of cliques containing v. All graphs considered will be assumed to be connected. A set S ⊆ V (G) is a uv-separator if vertices u and v are in diﬀerent connected components of G−S. It is minimal if no proper subset of S has the

1 Email: [email protected] aaURL: www.mate.unlp.edu.ar/~pdecaria 2 Email: [email protected] same property. S is a minimal vertex separator if there exist two nonadjacent vertices u and v such that S is a minimal uv-separator. S(G) will denote the family of all minimal vertex separators of G. Let F be a family of sets of vertices of a graph. F is separating if, for all v ∈ S F , T F = {v}. The intersection graph of F, L(F), has the members F ∈F v∈F of F as vertices, two of them being adjacent if and only if they are not disjoint. The clique graph K(G) is the graph L(C(G)). The two section of F, S(F), is a graph whose vertex set is S F , where u and v, u 6= v, are adjacent in F ∈F S(F) if and only if there exists F ∈ F such that {u, v} ⊆ F .

1.2 Basic chordal graphs and goals Chordal graphs were defined as those graphs for which every cycle of length greater than or equal to four has a chord. They can also be described in many other ways. A clique tree of G is a tree T whose vertex set is C(G) and such that for every v ∈ V (G), Cv induces a subtree of T . A graph is chordal if and only if it has a clique tree [4]. The clique graphs of chordal graphs, i.e., dually chordal graphs, also have a tree (the compatible tree) that characterizes them. A compatible tree of G is a spanning tree T of G such that every clique of G induces a subtree of T . Compatible trees are not hard to find. In fact, every clique tree of a chordal graph G is a compatible tree of K(G). However, it is not necessarily true that every compatible tree of K(G) is a clique tree of G. Basic chordal graphs appeared in this context [2] and were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. One of the aspects about basic chordal graphs that were studied is the recognition problem. Let S be a minimal vertex separator of a graph G. Denote by CS the family of cliques of G containing S and let BS be the family consisting of every clique of G that intersects every C ∈ C(G) such that C ∩ S 6= ∅. The following characterization enables a polynomial-time recognition of basic chordal graphs. Theorem 1.1 [2] A chordal graph G is basic chordal if and only if for all S ∈ S(G), BS = CS. Another important fact about basic chordal graphs is that, despite being a strict subclass of chordal graphs, their clique graphs are the same. Theorem 1.2 [2] The class of clique graphs of basic chordal graphs is equal to the class of dually chordal graphs. In this paper, we define and study some subclasses of basic chordal graphs. In Section 2, we introduce the class of hereditary basic chordal graphs, find several characterizations for them and show that this class is equivalent to some others, like strictly chordal graphs and (4,6)-leaf powers. We also show that the clique graphs of hereditary basic chordal graphs are the block graphs. In Section 3, we study the correspondence between special classes of clique trees and compatible trees, thus giving rise to basic DV and basic RDV graphs. We also characterize these classes and find their clique graphs.

2 Hereditary basic chordal graphs

A graph G is said to be hereditary basic chordal if G and all its induced subgraphs are basic chordal. These graphs clearly form a hereditary class and hence they have a family of minimal forbidden induced subgraphs. Other than the cycles Cn (n ≥ 4), the only two minimal forbidden induced subgraphs are the dart and the gem, as it will be shown later. As a consequence, we will ﬁrst explore some properties of gem-free graphs and of dart-free graphs.

Fig. 1. A dart (left) and a gem (right).

Proposition 2.1 Let G be a chordal graph. Then, G is gem-free if and only if every edge of K(G) is in some clique tree of G. Proposition 2.2 Let G be a chordal dart-free graph. Then, no minimal vertex separator of G contains another. A conjunction of the previous two propositions gives another characterization for the class of hereditary basic chordal graphs. Some other characterizations can also be found in the next result. Theorem 2.3 Let G be a chordal graph. The following are equivalent: (i) G is hereditary basic chordal. (ii) G is a (dart,gem)-free graph. (iii) Every edge of K(G) is in some clique tree of G and no minimal vertex separator of G contains another.

(iv) For every triple C1,C2,C3 of pairwise intersecting cliques of G, C1∩C2 = C1 ∩ C3 = C2 ∩ C3. (v) For all C ∈ C(G) and S ∈ S(G), S ∩ C 6= ∅ implies S ⊆ C. (vi) The minimal vertex separators of G are pairwise disjoint. The characterization of hereditary basic chordal graphs via minimal forbidden induced subgraphs reveals that this class appeared before in some other works under very different contexts and names. First, in 2005, William Kennedy studied strictly chordal graphs in his Master Thesis [3]. There, a graph G is said to be strictly chordal if G is chordal and for every subfamily F ⊆ C(G), I(F) := T C 6= ∅ and F = {C ∈ C∈F C(G): C ∩ I(F) 6= ∅} imply that C ∩ C0 = I(F) for all C,C0 ∈ F. Moreover, the thesis includes a proof of the fact that strictly chordal graphs are exactly the (dart-gem)-free chordal graphs. A connection with leaf powers was found later. Let G be a graph and k, l be integers such that 2 ≤ k < l. G is defined to be a (k,l)-leaf power if there exists a tree T whose set of leaves is V (G) and such that dT (u, v) ≤ k for all uv ∈ E(G) and dT (u, v) ≥ l for all uv 6∈ E(G)(dT denotes distance in T ). Brandstädt and Wagner [1] proved that strictly chordal graphs are exactly the (4,6)-leaf powers. Therefore, Theorem 2.4 Let G be a graph. The following are equivalent: (i) G is hereditary basic chordal. (ii) G is strictly chordal. (iii) G is a (4,6)-leaf power.

2.1 Clique graphs of hereditary basic chordal graphs We will show here that the class of clique graphs of hereditary basic chordal graphs is the class of block graphs. Note ﬁrst that hereditary basic chordal graphs are strongly chordal because the fact that they are gem-free implies that they are sun-free. Since the clique graph of a strongly chordal graphs is also strongly chordal, the clique graph of every hereditary basic chordal graph is chordal. In order to proceed, we need the following proposition: Proposition 2.5 Let G be a hereditary basic chordal graph and C,C0 be two cliques of G such that C ∩ C0 6= ∅. Then, the edge CC0 is contained in only one clique of K(G). The graphs whose edges are contained in only one clique are the diamond- free graphs. The class of chordal diamond-free graphs is equal to the class of block graphs. Therefore, the clique graphs of hereditary basic chordal graphs are block graphs. Furthermore, trees form a subclass of hereditary basic chordal graphs and their clique graphs are all block graphs. This outlines the proof of the initial claim of the section, which we now state as a theorem. Theorem 2.6 Let G be a graph. Then, G is the clique graph of a hereditary basic chordal graph if and only if G is a block graph.

3 Basic DV and basic RDV graphs

This section focuses on the existence of special types of clique trees and compatible trees and their connection with basic chordal graphs. Three subclasses of clique trees, described below, have been studied for more than twenty years. A UV clique tree of a graph G is a clique tree such that, for every v ∈ V (G), T [Cv] is a path of T .A DV clique tree of G is a clique tree such that its edges have been directed and, for every v ∈ V (G), T [Cv] is a directed path of T . An RDV clique tree of G is a DV clique tree that is rooted at a vertex w.A chordal graph is UV/DV/RDV if it has a UV/DV/RDV clique tree. The clique graphs of UV graphs are all dually chordal graphs. This and other reasons justify that UV graphs do not have a dual class. On the contrary, DV and RDV graphs do have each a dual class. The dually DV(RDV) graphs are the clique graphs of DV (RDV ) graphs. This duality is also reflected by the existence of characteristic trees. The DV(RDV) compatible tree is a (rooted) directed spanning tree with every clique of the graph inducing a directed path. We had mentioned before that every clique tree of a chordal graph G is compatible with K(G). There is a similar result for DV and RDV graphs. Proposition 3.1 Let G be a DV (RDV ) graph. Then, every DV (RDV ) clique tree of G is a DV (RDV ) compatible tree of K(G). Once again, the converse is not necessarily true. Given a DV (RDV ) graph, there could be a compatible DV (RDV ) tree of K(G) that is not a DV (RDV ) clique tree of G. We define a basic DV(RDV) graph as a DV (RDV ) basic chordal graph whose DV (RDV ) clique trees are the DV (RDV ) compatible trees of its clique graph. Basic DV and basic RDV graphs are not difficult to identify, as the following theorem shows. Theorem 3.2 Let G be a chordal graph. Then, (a) G is basic DV if and only if G is basic chordal and DV . (b) G is basic RDV if and only if G is basic chordal and RDV . The importance of Theorem 3.2 lies on showing that, for a basic chordal graph G, the correspondence is not only between the clique trees of G and the compatible trees of K(G), but also between the DV (RDV ) clique trees of G and the DV (RDV ) compatible trees of K(G), when they exist. Finally, we study the clique graphs of these classes. Given a dually chordal graph G, denote by SDC(G) the family of subsets of V (G) inducing a subtree of every compatible tree of G. If G is dually DV , define X (G) to be the family of sets of vertices of G inducing a directed path of every DV compatible tree of G. If G is dually RDV , Y(G) will denote the family of sets of vertices of G inducing a directed path of every RDV compatible tree of G. The connection between these families and clique graphs is the following: Theorem 3.3 Let H be a chordal graph. (a) Let G be dually DV . Then, K(H) = G and H is basic DV if and only if H is the intersection graph of a separating family F contained in SDC(G) ∩ X (G) and such that S(F) = G. (b) Let G be dually RDV . Then, K(H) = G and H is basic RDV if and only if H is the intersection graph of a separating family F contained in SDC(G) ∩ Y(G) and such that S(F) = G. One family F satisfying the conditions of Theorem 3.3 consists of the cliques of G and its unit sets of vertices. Thus, much like the clique graphs of basic chordal graphs are all dually chordal graphs, we have: Theorem 3.4 K(Basic DV ) = Dually DV and K(Basic RDV ) = Dually RDV .

References

[1] Brandst¨adtA. and P. Wagner, Characterising (k,l)-leaf powers, Discrete Applied Mathematics, 158 (2010), 110-122. [2] De Caria P. and M. Gutierrez, Comparing trees characteristic to chordal and dually chordal graphs, Electronic Notes in Discrete Math., 37 (2011), 33-38. [3] Keneddy W., “Strictly cordal graphs and phylogenetic roots,” Master Thesis, University of Alberta, 2005. [4] Shibata Y., On the tree representation of chordal graphs, J. Graph Theory, 12 (1988), 421-428. 4 Appendix

This section includes the details of the proofs of the propositions and theorems of the paper for the consideration of the referees. Proposition 2.1 Statement: Let G be a chordal graph. Then, G is gem-free if and only if every edge of K(G) is in some clique tree of G. Prerequisite knowledge: In order to follow the proof, the reader must be familiar with the concept of separating pair. Two cliques C and C0 of a graph G form a separating pair if C ∩ C0 is a minimal uv-separator for all u ∈ C \ C0 and v ∈ C0 \ C. Separating pairs are relevant in chordal graphs, as the following result shows: Proposition 4.1 [2] Let G be a chordal graph and CC0 be an edge of K(G). Then, CC0 is the edge of some clique tree of G if and only if C and C0 form a separating pair.

Proposition 4.1 will be used in many other proofs, sometimes implicitly. Proof of Proposition 2.1: Suppose that there exists an edge CC0 of K(G) that is in no clique tree of G. By Proposition 4.1, C,C0 is not a separating pair. Let v be a vertex in C ∩ C0 and P be a path of G − C ∩ C0 of minimum length among all the paths of G−C ∩C0 with initial vertex in C \S and ﬁnal vertex in C0 \ S.

If the length of P were 1, let v1, v2 be the vertices of P and v3, v4 be two 0 nonadjacent vertices, one in C and the other in C . Thus, {v, v1, v2, v3, v4} induces a gem.

If the length of P were 2, let v1, v2, v3 be the vertices of P , with v1 ∈ C 0 and v3 ∈ C . The cycle vv1v2v3v must have a chord and, by the construction of P , that chord is vv2. The deﬁnition of P also implies that v2 6∈ C, so we can take a vertex v4 ∈ C that is not adjacent to v2. v4 is not adjacent to v3, otherwise the deﬁnition of P would be contradicted. Thus, {v, v1, v2, v3, v4} induces a gem.

If the length of P were larger than 2, let v1, v2, v3, ..., vn be the vertices of 0 P , with v1 ∈ C and vn ∈ C . By the deﬁnition of P , the chords of the cycle vv1v2v3...vnv must have v as an endpoint, so we can infer that v is adjacent to all the vertices of P . Thus, {v, v1, v2, v3, v4} induces a gem. Therefore, G is not gem-free.

Conversely, suppose that G is not gem-free. Let v1, v2, v3, v4, v5 be vertices of G inducing a gem like the one of Figure 1, C be a clique of G containing 0 0 {v1, v2, v3} and C be another clique containing {v1, v4, v5}. Thus, CC is an 0 edge of K(G) because v1 ∈ C ∩ C but it is not a separating pair because 0 0 0 v2 ∈ C \ C ∩ C , v4 ∈ C \ C ∩ C and these two vertices are adjacent. Consequently, by Proposition 4.1, CC0 is the edge of no clique tree of G. Proposition 2.2 Statement: Let G be a chordal dart-free graph. Then, no minimal vertex separator of G contains another. Prerequisite knowledge: It is necessary to know that any minimal vertex separator of a chordal graph can be written as the intersection of a separating pair [2]. Proof of Proposition 2.2: Suppose on the contrary that there are two 0 0 distinct minimal vertex separators S and S such that S ⊆ S . Let v1 ∈ S, 0 v2 ∈ S \ S, C1,C2 and C3,C4 be separating pairs such that C1 ∩ C2 = S and 0 0 0 C3 ∩ C4 = S , v3 ∈ C3 \ S and v4 ∈ C4 \ S . The vertices v2, v3 and v4 are contained in the same connected component of G−S. We can assume without loss of generality that C1 \ S is contained in a connected component of G − S diﬀerent from that of {v2, v3, v4}. Let v5 ∈ C1 \ S. Thus, {v1, v2, v3, v4, v5} induces a dart, which is a contradiction. Therefore, no minimal vertex separator of G contains another. Theorem 2.3 Statement: Let G be a chordal graph. The following are equivalent: (i) G is hereditary basic chordal. (ii) G is a (dart,gem)-free graph. (iii) Every edge of K(G) is in some clique tree of G and no minimal vertex separator of G contains another.

Prerequisite knowledge: Given a tree T and u, v ∈ V (T ), T [u, v] denotes the vertices of the path in T from u to v. It is necessary to know the clique tree can also be deﬁned in this way: T is a clique tree of the chordal graph G if it is a spanning tree of K(G) and, for all C1,C2,C3 ∈ C(G), C3 ∈ T [C1,C2] implies that C1 ∩ C2 ⊆ C3. Proof of Theorem 2.3: (i) → (ii) It suﬃces to prove that the dart and the gem are not basic 0 chordal graphs. Let G be any of the two graphs of Figure 1, C1 = {v1, v2, v3}, C2 = {v1, v2, v4} and C3 be the clique containing {v1, v5}. Thus, the path 0 0 C1C3C2 is a compatible tree of K(G ) but not a clique tree of G . Therefore, the dart and the gem are not basic chordal graphs. (ii) → (iii) See Propositions 2.1 and 2.2.

(iii) → (iv) Let C1,C2,C3 be a triple of pairwise intersecting cliques of G. By (iii), there exists a clique tree T of G such that C1C2 ∈ E(T ). Suppose without loss of generality that C2 ∈ T [C1,C3]. As a consequence, C1 ∩ C3 ⊆ C1 ∩ C2 and C1 ∩ C3 ⊆ C2 ∩ C3. Since intersecting cliques are separating pairs, the intersection of the cliques of a separating pairs are minimal vertex separators and, by (iii), no minimal vertex separator contains another. We conclude that C1 ∩ C3 = C1 ∩ C2 and C1 ∩ C3 = C2 ∩ C3. (iv) → (v) Let S ∈ S(G) and C ∈ C(G) be such that S ∩ C 6= ∅. Take cliques C1 and C2 such that S = C1 ∩C2. If C = C1 or C = C2, then it is clear that S ⊆ C. If not, then C,C1,C2 is a triple of pairwise intersecting cliques. Therefore, by (iv), C ∩ C1 = C1 ∩ C2 = S, from which the inclusion S ⊆ C follows. (v) → (vi) Let S and S0 be minimal vertex separators of G such that S ∩ S0 6= ∅. We will prove that S = S0.

Let S be the intersection of two cliques C1 and C2 of G. The fact that 0 0 0 0 S ∩ S 6= ∅ implies that C1 ∩ S 6= ∅ and C2 ∩ S 6= ∅. By (v), S ⊆ C1 and 0 0 0 S ⊆ C2. Hence, S ⊆ C1 ∩ C2, that is, S ⊆ S. Similarly, we can prove that S ⊆ S0. Therefore, S = S0. (vi) → (i) Suppose that G is not hereditary basic chordal and let G0 be an induced subgraph of G that is not basic chordal. We need to prove that there exist two diﬀerent minimal vertex separators of G that are not disjoint. 0 Consider a minimal vertex separator S of G such that BS and CS, both 0 0 computed with respect to G , are diﬀerent. Let C, C1 and C2 be cliques of G such that C ∈ BS \CS, C1,C2 is a separating pair and C1 ∩ C2 = S. Next we will prove that C ∩ S 6= ∅.

As C ∈ BS, C ∩ C1 6= ∅ and C ∩ C2 6= ∅. C cannot intersect both C1 \ S and C2 \ S because otherwise C1 \ S and C2 \ S would not be separated by S, contradicting that C1,C2 is a separating pair. Thus, C∩C1 ⊆ S or C∩C2 ⊆ S. In either case, we conclude that C ∩ S 6= ∅. Let v1 be a vertex of C ∩ S. Since C 6∈ CS, we can also take a vertex v2 ∈ S \ C. There also exists a vertex v3 in C that is not adjacent to v2. v2 is clearly diﬀerent from v1 because S is complete. 0 0 0 Let S be a minimal v2v3-separator in G. Then, S 6= S because v2 ∈ S \ S 0 0 and S ∩ S 6= ∅ because v1 ∈ S ∩ S . Proposition 2.5 Statement: Let G be a hereditary basic chordal graph and C,C0 be two cliques of G such that C ∩ C0 6= ∅. Then, the edge CC0 is contained in only one clique of K(G). Proof: Let C00 be another clique of G adjacent to both C and C0 in K(G). Thus, C,C0,C00 is a triple of pairwise intersecting cliques and, by Theorem 2.3, C ∩ C00 = C ∩ C0. Therefore, C ∩ C0 ⊆ C00. We infer that the family of cliques of G containing C ∩ C0 form a complete set of K(G) that contains all the cliques adjacent to both C and C0 in K(G). Therefore, this family is the only clique of K(G) containing the edge CC0. Proposition 3.1 Statement: Let G be a DV (RDV ) graph. Then, every DV (RDV ) clique tree of G is a DV (RDV ) compatible tree of K(G). Prerequisite knowledge: It is necessary to know that a chordal graph without the 3-sun as an induced subgraph is a clique-Helly graphs [3]. Proof: Let T be a DV (RDV ) clique tree of G. Since the 3-sun is not a DV (RDV ) graph, it is the induced subgraph of no DV (RDV ) graph. Hence, G is a clique-Helly graph. This implies that every clique of K(G) is of the form Cv, for some v ∈ V (G). These sets induce directed paths of T because T is a DV (RDV ) clique tree. Therefore, T is a DV (RDV ) compatible tree. Theorem 3.2 Statement: Let G be a chordal graph. Then, (a) G is basic DV if and only if G is basic chordal and DV . (b) G is basic RDV if and only if G is basic chordal and RDV .

Proof: (a) Every basic DV graph is basic chordal and DV by deﬁnition. Conversely, suppose that G is basic chordal and DV . By Proposition 3.1, every DV clique tree of G is a DV compatible tree of K(G). So, in order to prove that G is basic DV , it remains to demonstrate that every DV compatible tree of K(G) is a DV clique tree of G. Let T be a DV compatible tree of K(G) and v be any vertex of G. Since G is basic chordal, T is a clique tree of G, so Cv induces a subtree of T . Furthermore, Cv is a complete set of K(G). Let D be a clique of K(G) containing Cv. As T is compatible with K(G), D induces a directed path of T . Hence, Cv induces a subtree of T contained in a directed path of T . Therefore, T [Cv] itself is a directed path. This completes the proof. (b) It is similar to the proof of (a). Theorem 3.3 Statement: Let H be a chordal graph. (a) Let G be dually DV . Then, K(H) = G and H is basic DV if and only if H is the intersection graph of a separating family F contained in SDC(G) ∩ X (G) and such that S(F) = G. (b) Let G be dually RDV . Then, K(H) = G and H is basic RDV if and only if H is the intersection graph of a separating family F contained in SDC(G) ∩ X (G) ∩ Y(G) and such that S(F) = G.

Prerequisite knowledge: We need this theorem:

Theorem 4.2 [1] Let G be a dually chordal graph and H be a chordal graph. Then, H is basic chordal and K(H) = G if and only if H is the intersection graph of a separating subfamily contained in SDC(G) whose two section is equal to G.

Proof of Theorem 3.3: (a) Suppose that H is basic DV and K(H) = G. We know that H is equal to the intersection graph of {Cv}v∈V (H). Every member of this family induces a subtree of every clique tree of H and a directed path of every DV clique tree of H. Since H is basic DV , this is equivalent to stating that every member of the family induces a subtree of every compatible tree of G and a directed path of every DV compatible tree of G. Thus, {Cv}v∈V (H) ⊆ SDC(G) ∩ X (G). It is not diﬃcult to verify that the two section of the family is K(H), which equals G, and that it is separating. Conversely, suppose that H is the intersection graph of a separating family F contained in SDC(G) ∩ X (G) and such that S(F) = G. Then, H can be seen as the intersection graph of directed paths of any DV compatible tree of G and hence H is DV . Furthermore, by Theorem 4.2, H is basic chordal and K(H) = G. Now we apply Theorem 3.2 to conclude that H is basic DV . (b) Similar to (a).

References

[1] P. De Caria, A joint study of chordal and dually chordal graphs, Ph. D. Thesis, Universidad Nacional de La Plata, 2012.

[2] M. Habib and J. Stacho, Reduced clique graphs of chordal graphs, European Journal of Combinatorics 33 (2012), 712-735.

[3] M. Gutierrez, Tree-Clique Graphs, Estudos e comunica¸c˜oesdo Instituto de Matem´atica,Universidade Federal do Rio de Janeiro 63 (1996), 7-26.