A Joint Study of Chordal and Dually Chordal Graphs

Universidad Nacional de La Plata

Facultad de Ciencias Exactas

Departamento de Matemática

A joint study of chordal and dually chordal graphs

Pablo De Caria

PhD. Thesis Advisor: Dr. Marisa Gutierrez

April, 2012

Celebrating the month of the 35th anniversary of my parents’ first meeting

In memory of aunt Sara

Thanks…

To my parents Francisco and Norma, for their support and company and because, as father and mother, they never stop surprising me in a favourable way.

To my sisters Laura and Mariana and my niece Ariadna, for all the time we shared and surely are to share together, and for being part of a family that I really cherish.

To Hernan and Paula, for their company, which helped me to get the energy to lead a better life. I look forward to sharing more adventures with you.

To María, for the many years of friendship, her permanent interest in my activities and her advice.

To Marisa, for making graphs known to me and letting me work on them with a level of freedom that made me feel very comfortable over all these years. I would also like to thank her for her words in the right moments and her guidance on several administrative issues.

To Juraj, for accepting to come and be a member of the jury and for his willingness, which was by no means limited to the evaluation of this thesis.

To Liliana and Oscar, the other two jurors, for their commitment to improving this work.

To Silvia, my official camerawoman at the symposiums, for being a very good mate.

To CONICET, for providing me with more than good conditions to focus on working on what I like, the fruit of which is this work.

III

Contents

Introduction 1

Deﬁnitions 3 List of deﬁnitions and notation ...... 6

1 Preliminaries 8 1.1 Chordal graphs ...... 8 1.1.1 Subclasses of chordal graphs ...... 11 1.2 Dually chordal graphs ...... 14 1.2.1 Subclasses of dually chordal graphs ...... 16

2 Special eccentric vertices in chordal graphs, dually chordal graphs and their subclasses 18

3 Minimal vertex separators of dually chordal graphs 25 3.1 Minimal vertex separators and compatible trees ...... 25 3.2 Minimal vertex separators and neighborhoods ...... 27 3.3 Characterizations ...... 31 3.4 Minimal vertex separators and strongly chordal graphs ...... 33

4 Basic chordal graphs 36 4.1 Subtree inducing sets and the concept of basis ...... 39 4.2 More results on basic chordal graphs and dually chordal graphs ...... 42 4.3 Determining possible sets of leaves for the compatible trees of a dually chordal graph ...... 50

5 Detecting the families of clique trees of chordal graphs 53 5.1 A detection method which involves counting ...... 53 5.2 Another detection method ...... 59 5.3 Finding all chordal graphs with a given family of clique trees ...... 62 5.4 Detecting the families of compatible trees of dually chordal graphs ...... 64

6 Conclusions 68

Bibliography 70

IV Introduction

Chordal graphs were originally defined as those graphs for which every cycle of length greater than or equal to four has a chord. Chordal graphs have been comprehensively studied because many applications have been found for them, especially in biology. As a result of those studies, a considerable number of new characterizations of chordal graphs arose, which involve diverse concepts like minimal vertex separators, simplicial vertices and clique trees. A clique of a graph G is a maximal set of pairwise adjacent vertices. The clique graph of G has the cliques of G as vertices, two of them being adjacent if and only if their intersection is not empty. A graph is said to be dually chordal if it is the clique graph of some chordal graph. Historically speaking, dually chordal graphs appeared more than twenty years ago in several independent researches under diverse names, like HT graphs, tree clique graphs and extended trees. Not only the names differed, but also the definitions were in different terms, which made dually chordal graphs have many characterizations. Some more years were necessary until it was discovered that all these definitions were equivalent. The results appearing in this work are numerous, but they mainly follow just two goals. As a first goal, new characterizations of the dually chordal graphs were sought in a way that they extend some others that were already known. We will see this in Chapter 3 and, to a lesser degree, in Chapter 4. As a second goal, given that many characterizations of chordal and dually chordal graphs are related, the similarities are exploited to perform a joint study of both classes that explores their structure and approaches some deterministic problems. We will see this, to a greater or lesser extent, in Chapters 2, 4 and 5. In Chapter 1, the basic properties of chordal and dually chordal graphs are reviewed. The class of chordal graphs has several subclasses that will turn out to be interesting for us, like power chordal, doubly chordal and strongly chordal graphs. All these classes, in conjunction with dually chordal graphs, are characterized by the existence of special vertices: simplicial vertices for chordal graphs, vertices with a maximum neighbor for dually chordal graphs, power- simplicial vertices for power chordal graphs, doubly simplicial vertices for doubly chordal graphs and simple vertices for strongly chordal graphs. It was known that a chordal graph either is complete or it possesses two nonadjacent simplicial vertices. In Chapter 2, we will see that similar properties concerning the other classes and their characteristic vertices are true. It will also be possible to prove some other common metric properties for these classes. In Chapter 3, new characterizations of dually chordal graphs in terms of their minimal vertex separators are obtained. Besides, a more complete study of minimal vertex separators of dually chordal graphs is conducted, where they are related to neighborhoods, among other things. Both chordal and dually chordal graphs possess characteristic trees. A clique tree of a chordal graph is a tree whose vertices are the cliques of the graph and such that, for every vertex v,

1 the set Cv of cliques containing v induces a subtree in the tree. A compatible tree of a dually chordal graph is a spanning tree such that every clique induces a subtree. It is possible to relate clique trees and compatible trees via clique graphs. In fact, every clique tree of a chordal graph is a compatible tree of its clique graph. However, the converse is not true for all chordal graphs. The chordal graphs for which the converse is true are called basic chordal graphs. In Chapter 4, some characterizations of basic chordal graphs are found. All the tools that are used to ﬁnd the characterizations are important in themselves and some applications of them will be shown. For example, it will be shown how several problems about the compatible trees of a dually chordal graph can be transformed into problems about the clique trees of a chordal graph, the latter having been more studied before. Finally, in Chapter 5, the following question is answered: given a family T of trees, determine whether T is the family of all clique trees of some chordal graph. If so, all chordal graphs having T as the family of clique trees are characterized. The question whether T is the family of all compatible trees of some dually chordal graph will also be discussed.

2 Deﬁnitions

A graph G is a triple consisting of a vertex set V (G), an edge set E(G) and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints. G is finite if both V (G) and E(G) are finite sets. G is simple if every edge of G has two distinct endpoints, and no two edges have exactly the same two endpoints. This work deals with finite simple graphs only. Given u, v ∈ V (G), uv will denote the edge whose endpoints are u and v. Two distinct vertices are adjacent when they are the two endpoints of some edge of the graph. G + uv is the graph such that V (G + uv) = V (G) and E(G + uv) = E(G) ∪ {uv}; and G − uv is the graph such that V (G − uv) = V (G) and E(G − uv) = E(G) \{uv}. The complement of G, or G, is the graph with the same vertices as G and such that, for all u, v ∈ V (G), uv ∈ E(G) if and only if u 6= v and uv 6∈ E(G). A graph G0 is a subgraph of G if V (G0) ⊆ V (G) and E(G0) ⊆ E(G). The subgraph induced by A ⊆ V (G), or G[A], has A as vertex set and two vertices are adjacent in G[A] if and only if they are adjacent in G. The graph G − A is defined as G − A = G[V (G) \ A]. We say that G0 is an induced subgraph of G if there exists V 0 ⊆ V (G) such that G0 = G[V 0]. G is a complete graph if all of its vertices are pairwise adjacent. A subset of V (G) is complete if it induces a complete subgraph of G.A clique is a maximal complete set. The family of cliques of G is denoted by C(G). Given v ∈ V (G), the open neighborhood of v in G, or NG(v), is the set of all the vertices adjacent to v. The closed neighborhood of v in G, or NG[v], is defined as NG[v] = NG(v) ∪ {v}. It will be usual to use the notation N(v) and N[v] when it is clear what graph we are working on. This remark also holds for many other concepts that are going to be defined below. The degree of v in G, or degG(v), is the number of vertices to which v is adjacent in G, that is, degG(v) = |NG(v)|. We say that v is universal if it is adjacent to every other vertex in G, i.e., NG[v] = V (G). It is simplicial if NG[v] is complete. This is equivalent to NG[v] being a clique. Any clique that is equal to the closed neighborhood of a vertex is called a simplicial clique. If u and v are two distinct vertices such that NG[u] ⊆ NG[v], then we say that v dominates u. A walk of G is a sequence v1v2...vn of vertices such that consecutive vertices are adjacent in G. The length of the walk is n − 1. If all the vertices of the walk are different, then we say that it is a path. If the only identical vertices are the first and the last one, then we say that it is a cycle.A chord of a cycle is an edge whose endpoints are two nonconsecutive vertices of the cycle. G is a connected graph if, for all u, v ∈ V (G), G has a path beginning at u and ending at v. Such a path is called a uv-path. Otherwise, G is disconnected.A connected component of G is a maximal connected subgraph of G. It is clear that, if G is connected, there is only one connected component, which is G itself. Let G be connected and u and v be two nonadjacent vertices of G.A uv-separator is a set S ⊆ V (G) such that u and v are in different connected components of G − S. The separator is

3 minimal if no proper subset of S has the same property. There are some important properties of uv-minimal separators that are worthwhile noting. If S is a minimal uv-separator, then, for every s ∈ S, there must exist a uv-path such that s is its only vertex in S; otherwise, S \{s} would also be a uv-separator, contradicting the minimality of S. The vertex preceding s in that path is in the same connected component of G − S as u, and the vertex following it is in the same connected component of G − S as v. Therefore, every vertex of a minimal uv-separator is adjacent to at least one vertex of the connected component of each vertex that it separates. We will just use the term minimal vertex separator to refer to a set separating some pair of nonadjacent vertices which is minimal in that sense. The family of minimal vertex separators of G will be denoted by S(G). The distance between u and v in G, or dG(u, v), is the length of a shortest uv-path of G. If there is no uv-path in G, then dG(u, v) is defined to be infinity. The disk centered at v with k radius k, or NG[v], is the set of vertices at distance at most k from v. The family of disks of G is denoted by D(G). The kth-power of G, or Gk, is another graph which has the same vertices as G, where two of them are adjacent in Gk if and only if the distance in G between them is at k most k. In other words, for every v ∈ V (G), NGk [v] = NG[v]. The eccentricity of v ∈ V (G) is eccG(v) = max{dG(v, w), w ∈ V (G)}. A vertex w is eccentric to v if no vertex of G is further away from v than w, that is, if eccG(v) = dG(v, w). Another important concept related to the distance in graphs is the diameter. It is defined as the maximum possible distance between two vertices of the graph, i.e., diam(G) = max{dG(v, w): v, w ∈ V (G)}. Two graphs G and G0 are isomorphic if there exists a bijective function f : V (G) → V (G0) such that, for all u, v ∈ V (G), uv ∈ E(G) if and only if f(u)f(v) ∈ E(G0). Such a function is called an isomorphism. We do not distinguish isomorphic graphs. A graph T is a tree if it is connected and without cycles. A connected subgraph of a tree is given the name of subtree.A leaf of a tree is a vertex of degree one. T is a spanning tree of G if it is a tree such that T is a subgraph of G and V (T ) = V (G). Trees have many characterizations. In fact, the following are equivalent:

• T is a tree.

• T is connected and |E(T )| = |V (T )| − 1.

• T has no cycles and |E(T )| = |V (T )| − 1.

• For u, v ∈ V (T ), T has a unique uv-path.

With regard to the last item, that path is denoted by T hu, vi. The set of vertices of that path is denoted by T [u, v] and T (u, v) := T [u, v] \{u, v}. If T is a family of trees, all with the same vertex set, and u and v are two vertices, define T [u, v] = S T [u, v]. T ∈T We use the word class to refer to a family of graphs. A graph class is hereditary if every induced subgraph of a graph of the class is also in the class. Every hereditary class of graphs can be characterized by the existence of a family of minimal forbidden induced subgraphs, i.e., a family such that each graph of it is not in the class and every of its induced subgraphs (different from the graph itself) is in the class. However, it is usually difficult to find the family of minimal forbidden induced subgraphs of a hereditary graph class. Let P be a graph property. The class of hereditary P graphs consists of all the graphs G such that G and every of its induced subgraphs have the property P . Thus defined, this is clearly a hereditary class.

4 Figure 1: For the family of trees in the ﬁgure, T [2, 3] = {1, 2, 3}.

Let F be a family of nonempty sets. If F ∈ F, F is called a member of F. If v ∈ S F , we F ∈F say that v is an element of the family. F is intersecting if the intersection of every two members of F is not empty. F is Helly if, for every intersecting subfamily F 0 of F, the intersection of all the members of F 0 is not empty. If C(G) is a Helly family, then we say that G is a clique-Helly graph. Call F separating if, for every two distinct elements v and w, there exists F ∈ F such that v ∈ F and w 6∈ F . The intersection graph of F, or L(F), has the members of F as vertices, where two of them are adjacent if and only if they are not disjoint. Although it is an abuse of language, the term intersection graph will also be applied to families of subtrees of a tree. In that case, it is said that two subtrees intersect when they have a common vertex. The clique graph of G, or K(G), is the intersection graph of C(G). The function K, having the set of all graphs as domain and codomain, which assigns to each graph its clique graph is given the name of clique operator. The two section of F, or S(F), is another graph whose vertices are the elements of F, where two of them are adjacent if and only if there exists a member of F to which both belong. The dual of the family F, or DF, is another family consisting of the sets Dv = {F ∈ F : v ∈ F }, where v is an element of F. For the particular case of C(G), Cv will be used instead of Dv. A more general notation will also be used: given a set A of vertices, CA := {C ∈ C(G): A ⊆ C}.

Figure 2: The intersection graph of the family {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}.

Two families F and F 0 are isomorphic if there exist two bijections a : F → F 0 and b : S F → S F such that, for every F ∈ F, a(F ) = b(F ). We will consider the equality F ∈F F ∈F0 of families and the isomorphism of families as equivalent concepts.

5 List of deﬁnitions and notation

Adjacent vertices, 3 Distance, 4 Dominated vertex, 3 Basic chordal graph, 36 Doubly chordal graph, 16 Basis, 40 Doubly perfect ordering, 16 BF , 47 Doubly simplicial vertex, 16 BS, 47 Dual dimension, 42 Dual family, 5 C( ), 3 Dual leafage, 51 CA, CS, 5 Dually chordal graph, 14 Chord, 3 Dually DV graph, 17 Chordal graph, 8 Dually RDV graph, 17 Ch(T ), 64 Duv, 61 Graph class, 4 Dv, 5 Clique, 3 DV clique tree, 11 Clique graph, 5 DV compatible tree, 17 Clique-Helly graph, 5 DV graph, 11 Clique operator, 5 D(v, i), 29 Clique tree, 10 Closed neighborhood, 3 ecc( ), eccG( ), 4 Compatible tree, 15 Eccentricity, 4 Complement of a graph, 3 Eccentric vertex, 4 Complete graph, 3 Edge, 3 Complete set, 3 E(G), 3 Connected component, 3 Element, 5 Connected graph, 3 Endpoints, 3 Connected positive boolean set, 46 Equivalence class, 53 Connected union, 40 Equivalence relation, 53 Correspondence, 43 E(T ), 56 Cv, 5 Cycle, 3 Finite graph, 3 Forbidden subgraph, 4 D( ), 4 d(, ), dG(, ), 4 G, 3 deg( ), degG( ), 3 G[ ], 3 Degree, 3 G − A, G − v, G − S, 3 DF, DC(G), 5 Generating family, 40 diam( ), 4 Gi, 8 Diameter, 4 Gk, 4 Digraph, 11 Graph, 3 Dimension, 42 G − uv, T − uv, 3 Directed graph, 11 G + uv, T + uv, 3 Directed path, 11 Disconnected graph, 3 Helly family, 5 Disk, 4 Hereditary class, 4

6 Hereditary P graph, 4 Root, 11 HT , 56 Rooted tree, 11 ∗ G HT , 56 RS , 53 HT,S, 54 S( ), 5 Induced subgraph, 3 S( ), 4 Intersecting family, 5 S-admissible set, 54 Intersection graph, 5 SC( ), 39 Isomorphic, 4, 5 SDC( ), 40 Isomorphism, 4 Separating family, 5 Separating pair, 37 K( ), 5 uv-separator, 3 Simple elimination ordering, 13 L( ), 5 Simple graph, 3 L( ), 50 Simple vertex, 13 Leaf, 4 Simplicial clique, 3 Leafage, 51 Simplicial vertex, 3 Spanning tree, 4 Maximum neighbor, 14 Span of F, 64 Maximum neighborhood ordering, 14 Sp(F), 64 Member, 5 s(T ), 32 Minimal separator, 4 Strong chord, 13 Strongly chordal graph, 13 Subgraph, 3 N( ), N ( ), 3 G Subtree, 4 N[ ], NG[ ], 3 k k Sun, 12 N [ ], NG[ ], 4 nK , 46 2 τ ( ), 10 T ( , ), 4 Open neighborhood, 3 T [ , ], 4 T [ , ], 4 Path, 3 T h , i, 4 uv-path, 3 T [C ,C ], 60 Perfect elimination ordering, 8 G 1 2 0 Tree, 4 P (H ), 56 Two section, 5 Positive boolean set, 46 Power chordal graph, 12 Universal vertex, 3 Power of a graph, 4 UV clique tree, 11 Power perfect ordering, 13 UV graph, 11 Power simplicial vertex, 13 V ( ), 3 Quotient set, 53 Vertex, 3

RDV clique tree, 11 Walk, 3 RDV compatible tree, 17 Wheel, 17 RDV graph, 11 T Wn, 17 RH0 , 58

7 Chapter 1

Preliminaries

The purpose of this chapter is to show all the previous knowledge that is required for the development of this work. The basic properties of chordal graphs and dually chordal graphs will be stated here.

1.1 Chordal graphs

Chordal graphs are defined as those for which every cycle of length greater than or equal to four has a chord. It is clear from this definition that the class of chordal graphs is hereditary. Chordal graphs have many practical applications, especially in biology. One good example of this are phylogenetic trees [19, 20], used to model the evolutionary history of species, proteins, etc. However, this section is focused on the characterizations that chordal graphs have, whose details are given below. The first one is due to Dirac and involves minimal vertex separators. Theorem 1.1. [4] Let G be a graph. Then, G is chordal if and only if every minimal vertex separator of G is complete. Recall that a vertex v of a graph G is simplicial if N[v] is complete in G. Dirac was also able to use the previous characterization to prove that every chordal graph has a simplicial vertex. Theorem 1.2. [4] Let G be a chordal graph. Then, G has a simplicial vertex. If G is not complete, then it has two nonadjacent simplicial vertices. Let v be a simplicial vertex of the chordal graph G. Since the class of chordal graphs is hereditary, G − v is also chordal. If G − v has at least one vertex, then we can apply Theorem 1.2 again to conclude that it has a simplicial vertex w. Now we could consider the chordal graph G − {v, w} and repeat the procedure as many times as possible until no vertex is left. This motivates the introduction of the concept of perfect elimination ordering. An ordering v1v2...vn of the vertices of G is a perfect elimination ordering if, for all 1 ≤ i ≤ n, vi is simplicial in Gi := G[{vi, ..., vn}]. The second characterization of chordal graphs is as follows:

8 Theorem 1.3. [10] Let G be a graph. Then, G is chordal if and only if it has a perfect elimination ordering. It is clear from the remark after Theorem 1.2 that every chordal graph has a perfect elimination ordering. It is not diﬃcult to prove the converse. Let v1v2...vn be a perfect elimination ordering of G and C be a cycle of G of length greater than or equal to four. Let vi be the ﬁrst vertex of the ordering which is in C. Then, as C is also a cycle of Gi and vi is simplicial in this graph, the two vertices adjacent to vi in C induce a chord. The next characterizations of chordal graphs will show that they can be represented by using trees. Theorem 1.4. [11] Let G be a graph. Then, G is chordal if and only if G is the intersection graph of a family of subtrees of a tree T . An example of a chordal graph and a representation of it as the intersection graph of a family of subtrees of a tree can be found in Figure 1.1.

Figure 1.1: A chordal graph and a representation of it as the intersection graph of a family of subtrees of a tree.

The tree T in the characterization is not unique. In fact, new vertices could be added to it to get a larger tree. If the family of subtrees is not modiﬁed, then the intersection graph is the same chordal graph. However, it is advisable to have a tree with minimum number of vertices. Let {Ti}1≤i≤n be a family of subtrees of T and assume that its intersection graph is connected. For every vertex v ∈ V (T ) such that, for all 1 ≤ i ≤ n, v 6∈ V (Ti), remove v from T . What remains from T is a subtree of it because we assumed that the intersection graph 0 0 is connected. Call that subtree T . Then, {Ti}1≤i≤n is also a family of subtrees of T and its intersection graph is still the same. There is a second procedure to reduce the number of vertices of T . Suppose that there exists and edge xy in T such that every subtree of the family that has x as a vertex also has y as a vertex. Then, we contract the edge xy into y. More precisely, we deﬁne a 00 00 new tree T such that V (T ) = V (T ) \{x}, NT 00 [y] = (NT [x] ∪ NT [y]) \{x} and, for

9 00 z ∈ V (T ) \{y}, NT 00 [z] = NT [z] if x 6∈ NT [z] and NT 00 [z] = (NT [z] ∪ {y}) \{x} if x ∈ NT [z]. If we also contract the edge xy in the subtrees of the family that contain x, it is not difficult to see that we obtain a family of subtrees of T 00 whose intersection graph is still the same graph. Suppose that we removed as many vertices and contracted as many edges in T as possible. The subtrees of the family containing a given vertex x clearly form a complete set of the intersection graph. What is more, this set is a clique and that there is a one to one correspondence between the cliques of the intersection graph and the vertices of the tree. Thus, for each vertex of the intersection graph, the set of cliques containing it induces a subtree, namely, the subtree of the family that represents it. A clique tree of a connected graph G is a tree T such that V (T ) = C(G) and, for every v ∈ V (G), the set Cv of cliques of G that contain v induces a subtree of T . Clique trees can also be defined differently: Proposition 1.5. Let G be a graph and T be a tree such that V (T ) = C(G). The following are equivalent:

(a) T is a clique tree of G.

(b) For all C1,C2,C3 ∈ C(G), C3 ∈ T [C1,C2] implies C1 ∩ C2 ⊆ C3.

Proof. (a) ⇒ (b). Let C1,C2,C3 ∈ C(G) be such that C3 ∈ T [C1,C2] and v be a vertex of C1 ∩ C2. Then, C1 and C2 are in Cv. Since Cv induces a subtree of T and C3 ∈ T [C1,C2], C3 ∈ Cv, that is, v ∈ C3. Therefore, every element of C1 ∩ C2 is an element of C3 and the inclusion follows. (b) ⇒ (a). Let v be a vertex of G and C1, C2 be cliques in Cv. Then, v ∈ C1 ∩ C2. Suppose that C3 ∈ T [C1,C2]. Then, C1 ∩ C2 ⊆ C3 and hence v ∈ C3, i.e., C3 ∈ Cv. Therefore, T [Cv] is a connected subgraph of T , that is, Cv induces a subtree of T . Clique trees can also be used to characterize chordal graphs. Theorem 1.6. [11] Let G be a connected graph. Then, G is chordal if and only if G has a clique tree. Sketch of proof. Suppose that G is chordal. Then, by Theorem 1.4, there exists a tree T such that G is the intersection graph of a family of subtrees of T . Use the ideas discussed after Theorem 1.4 to obtain a clique tree from T . Conversely, let T be a clique tree of G. Then, G is the intersection graph of {T [Cv]}v∈V (G). Therefore, by Theorem 1.4, G is chordal.

The family of clique trees of a chordal graph G will be denoted by τ (G). The clique tree is the model of tree representation for chordal graphs that is going to be used the most usually. One reason for this is the fact that many structural properties of the chordal graphs are clearly visible in their clique trees, as we shall see later.

10 Figure 1.2: After discarding the redundant vertices of the tree of the representation of Figure 1.1, a clique tree is obtained.

1.1.1 Subclasses of chordal graphs UV, DV and RDV graphs It is easy to define subclasses of the chordal graphs by imposing restrictions to the trees and families of subtrees used to represent them as intersection graphs. A directed graph or digraph is a graph G such that, for every edge of G, an initial endpoint and a final endpoint is defined for it. If e is an edge of G whose initial endpoint is u and whose final endpoint is v, then we say that e is oriented from u to v. A path v1v2...vn of a directed graph is a directed path if, for all 1 ≤ i ≤ n − 1, the edge vivi+1 is oriented from vi to vi+1. A directed tree T is said to be rooted if there exists a vertex v such that, for every w ∈ V (T ), T hv, wi is a directed vw-path. Such a vertex v is called the root. Let G be a chordal graph. G is a UV graph if it can be represented as the intersection graph of a family of paths of a tree. G is a DV graph if it can be represented as the intersection graph of a family of directed paths of a directed tree. G is an RDV graph if it can be represented as the intersection graph of a family of directed paths of a rooted directed tree. The three classes are clearly hereditary, where RDV graphs is a subclass of DV graphs, and DV graphs is a subclass of UV graphs. Given the way that UV , DV and RDV graphs were defined, they can also be characterized by using clique trees. Theorem 1.7. [21] Let G be a connected graph. Then, (1) G is UV if and only if there exists a tree T such that V (T ) = C(G) and, for every v ∈ V (G), Cv induces a path of T . (2) G is DV if and only if there exists a directed tree T such that V (T ) = C(G) and, for every v ∈ V (G), Cv induces a directed path of T . (3) G is RDV if and only if there exists a rooted directed tree T such that V (T ) = C(G) and, for every v ∈ V (G), Cv induces a directed path of T . A tree like the one mentioned in (1) is called a UV clique tree. Similarly, the trees of (2) and (3) are called DV clique tree and RDV clique tree, respectively.

11 Now we take advantage of Theorem 1.7 to show that chordal graphs, UV graphs, DV graphs and RDV graphs are all different classes. A k-sun is defined, for all k ≥ 3, as a graph with vertex set {v1, v2, ..., vk, w1, w2, ..., wk} such that {v1, v2, ..., vk} is complete, N[wi] = {vi, vi+1, wi}, i = 1, 2, ..., k − 1, and N[wk] = {v1, vk, wk}. An even sun is a sun such that k is even. An odd sun can be similarly defined.

Figure 1.3: A 3-sun and a 4-sun.

The suns are chordal graphs. The family of cliques of a k-sun has k+1 members C,C1, ..., Ck, where C = {v1, v2, ..., vk} and Ci = N[wi], i = 1, 2, ..., k. We now prove that the k-sun has a unique clique tree.

Let T be a clique tree of the k-sun. Since, by deﬁnition of clique tree, every Cvi induces a subtree of T , Cvi ∩ Cvi+1 , i = 1, ..., k − 1, and Cv1 ∩ Cvk each induce a subtree of T as well. Thus, CCi is an edge of T , i = 1, ..., k. We conclude that the only clique tree of the k-sun is the one where C is a universal vertex. It is simple to see that this tree is also a UV clique tree. Therefore, suns are UV graphs. However, the 3-sun is not DV because its clique tree cannot be properly directed to obtain a DV clique tree. More generally, no odd sun is a DV graph. However, it is not diﬃcult to verify that the even suns are DV graphs. Now we prove that the 4-sun is not an RDV graph. Let T be the clique tree of the 4-sun. We need to demonstrate that T cannot be transformed into an RDV clique tree.

The root of T cannot be C because in that case Cvi would not induce a directed path of T , i = 1, 2, 3, 4.

Nor can the root be C1 because otherwise neither Cv3 nor Cv4 would induce a directed path of T . Similarly, Ci cannot be a root, i = 2, 3, 4. Therefore, T cannot be rooted to form an RDV clique tree and hence the 4-sun is not an RDV graph. More generally, no sun is an RDV graph. It remains to ﬁnd a diﬀerence between chordal graphs and UV graphs. Let G be a graph such that V (G) = V 0 ∪ {v}, where V 0 is a set inducing a 3-sun in G and v is a universal vertex of G. Thus, G has the same number of cliques as a 3-sun and these graphs also have the same clique tree. Thus, Cv does not induce a path of the only clique tree of G. Therefore, G is a chordal graph that is not UV .

Power chordal graphs A graph G is power chordal if G and all of its powers are chordal. This deﬁnition can be simpliﬁed.

12 Proposition 1.8. [7] Let G be a graph and k be a natural number such that Gk is chordal. Then, Gk+2 is also chordal. We can infer from Proposition 1.8 that G is power chordal if and only if G and G2 are chordal. Then, there exist a perfect elimination ordering for G and a perfect elimination ordering for G2. It is even true that the orderings can be chosen to be the same. A vertex v of G is power simplicial if it is simplicial in both G and G2. Power chordal graphs satisfy a property which is very similar to Theorem 1.2. Theorem 1.9. [3] Let G be a power chordal graph. Then, G is complete or G has two nonadjacent power simplicial vertices. Theorem 1.2 made it possible to prove the existence of perfect elimination orderings in chordal graphs. Theorem 1.9 can be used similarly. An ordering v1v2...vn of the vertices of G is a power perfect ordering if, for all 1 ≤ i ≤ n, vi is power simplicial in Gi. Then: Theorem 1.10. [3] Let G be a graph. The following are equivalent: • G is power chordal.

• G and G2 are chordal.

• G has a power perfect ordering.

Strongly chordal graphs Strongly chordal graphs are defined as those chordal graphs for which every cycle whose length is even and greater than or equal to six has a chord joining two vertices at an odd distance in the cycle. Such a chord is called a strong chord. We see from this definition that the class of strongly chordal graphs is hereditary. Like chordal graphs, it can also be characterized in terms of elimination orderings. A vertex v of a graph G is simple if the set {N[u]: u ∈ N[v]} is totally ordered by inclusion. Simple vertices are also simplicial. To prove it, let u1 and u2 be two vertices in N[v]. Then, N[u1] ⊆ N[u2] or N[u2] ⊆ N[u1]. We infer that u1 ∈ N[u2] or u2 ∈ N[u1]. In either case, u1 and u2 are neighbors. Therefore, N[v] is a complete set and hence v is simplicial. A linear ordering v1v2...vn of V (G) is called a simple elimination ordering if, for all 1 ≤ i ≤ n, vi is simple in Gi. Then, by the previous paragraph, a simple elimination ordering is an special type of perfect elimination ordering. Then, the first characterization of strongly chordal graphs that is written is the following. Theorem 1.11. [8] A graph G is strongly chordal if and only if it has a simple elimination ordering. As a hereditary class, strongly chordal graphs can also be characterized by minimal forbidden induced subgraphs. Of all the graphs we have considered so far, the suns are an example of chordal graphs that are not strongly chordal. If we name the vertices of a k-sun as in the definition, then v1w1v2w2...viwi...vkwkv1 is an even cycle of length at least six without strong chords. Alter- natively, we can see that the suns are not strongly chordal because none of them have simple vertices. All this being said, the next characterization to be given is: Theorem 1.12. [8] A graph G is strongly chordal if and only if it is chordal and without induced suns.

13 1.2 Dually chordal graphs

A graph is dually chordal if it is the clique graph of a chordal graph. Unlike chordal graphs, which are initially defined in terms of cycles and chords in most books and courses, dually chordal graphs do not have a standard definition. As it was said in the introduction, they arose in many guises in the pioneering studies and, as a result, no definition of a dually chordal graph became prevalent. The usage of the definitions is mainly dependent on the purposes of the investigations concerning them. The decision in this work to start with the definition involving clique graphs has no special reason. It is just based on the fact that it does not require the introduction of new concepts to be stated. We start this section by discussing the connection between dually chordal graphs and linear orderings. Let v and w be vertices of a graph G. We say that w is a maximum neighbor of v when N 2[v] ⊆ N[w]. The case that v and w are equal is not excluded for this definition. If v is a maximum neighbor of itself, then every vertex at distance at most two from v is at distance at most one from v. Thus, in case that G is connected, v is a universal vertex. An ordering v1v2...vn of the vertices of G is a maximum neighborhood ordering if, for all 1 ≤ i ≤ n, vi has a maximum neighbor in Gi. This definition allows to state the following characterization of dually chordal graphs. Theorem 1.13. [3] Let G be a graph. Then, G is dually chordal if and only if it has at least one maximum neighborhood ordering.

Figure 1.4: v1v7v6v2v3v5v4 is a maximum neighborhood ordering of G. In the ﬁgure, the steps to obtain that ordering are shown. The light blue vertices are the ones that are added to the ordering at each step and the green vertices are their maximum neighbors. The set of colored vertices is equal to the disk centered at the light blue vertex with radius 2. The violet edges help to verify that the green vertex is indeed a maximum neighbor.

It is easy to prove that some graphs are not dually chordal by using this characterization. For example, the cycle of length four is not dually chordal because it has no vertex with a maximum neighbor. The same can be said about the suns. However, the graph consisting of an induced cycle of length four plus a universal vertex is dually chordal.

14 More generally, every graph with a universal vertex u is dually chordal. In fact, every vertex has u as a maximum neighbor. Thus, any ordering of the vertices of the graph ending in u is a maximum neighborhood ordering. As a consequence of the previous paragraphs, we see that the class of dually chordal graphs is not hereditary. Now we will see that dually chordal graphs have associated structures analogous to the clique trees of chordal graphs. Theorem 1.14. [3] Let G be a connected graph. The following are equivalent:

1. G is dually chordal.

2. G has a spanning tree T such that every clique of G induces a subtree of T.

3. G has a spanning tree T such that the closed neighborhood of every vertex of G induces a subtree of T.

4. G has a spanning tree T such that every disk of G induces a subtree of T.

It is even true that every spanning tree fulﬁlling 2., 3. or 4. also fulﬁlls the other two. Such a tree will be said to be compatible with G.

Figure 1.5: A chordal graph G and its clique graph. The edges of a tree compatible with K(G) are in color. Yellow is used to indicate the subtree induced by NK(G)[C8].

The characterization of connected dually chordal graphs given by 3. can be rephrased as follows: Theorem 1.15. [13] A connected graph G is dually chordal if and only if there exists a spanning tree T of G such that, for all xy ∈ E(G) and z in the xy-path of T , xz ∈ E(G) and yz ∈ E(G).

Now we are in a better position to understand why dually chordal graphs are called that way. Theorem 1.14 implies that a graph G is dually chordal if and only if C(G) can be represented as a family of subtrees of a tree. On the other side, Theorem 1.6 implies that G is chordal if and only if the dual of C(G) can be represented as a family of subtrees of a tree.

15 Finally, we include another characterization of the dually chordal graphs. This one also involves cliques.

Theorem 1.16. [6] A graph G is dually chordal if and only if G is clique-Helly and K(G) is chordal.

1.2.1 Subclasses of dually chordal graphs Doubly chordal graphs A graph G is doubly chordal if it is both chordal and dually chordal. It is clear that a doubly chordal graph has a perfect elimination ordering and also has a maximum neighborhood ordering. A vertex v is doubly simplicial if it is simplicial and has a maximum neighbor. A linear ordering v1v2...vn of the vertices of G is doubly perfect if, for all 1 ≤ i ≤ n, vi is doubly simplicial in Gi. We can use such an ordering to characterize doubly chordal graphs. Theorem 1.17. [3] A graph G is doubly chordal if and only if G has a doubly perfect ordering.

Strongly chordal graphs Let G be a graph and v be a simple vertex of it. By deﬁnition, we know that the set {N[u]: u ∈ N[v]} is totally ordered by inclusion. Therefore, there exists a vertex w such that N[u] ⊆ N[w] for all u ∈ N[v]. Thus, w is a maximum neighbor of v. We conclude through this kind of reasoning that every simple elimination ordering is a maximum neighborhood ordering. Therefore, strongly chordal graphs form a subclass of dually chordal graphs. What is more, every simple elimination ordering is doubly perfect, so the strongly chordal graphs are contained in the class of doubly chordal graphs. There is an even stronger connection between strongly chordal graphs and dually chordal graphs. We get to it with the help of two propositions.

Proposition 1.18. [13] Let G be a chordal graph that is not clique-Helly. Then, G has an induced 3-sun.

Proposition 1.19. [13] Let G be a chordal clique-Helly graph such that K(G) is not chordal. Then, there exists a number k ≥ 4 such that G has an induced k-sun.

In view of Theorem 1.16 and the characterization of strongly chordal graphs by forbidden induced subgraphs, we conclude from the last two propositions that every induced subgraph of a strongly chordal graph is dually chordal. Thus, every strongly chordal graph is a hereditary dually chordal graph. Furthermore, the cycles of length greater than or equal to four and the suns are not dually chordal, so none of them can be an induced subgraph of a hereditary dually chordal graph. Thus, we have the following characterization:

Theorem 1.20. A graph G is strongly chordal if and only if it is a hereditary dually chordal graph.

16 Dually DV and dually RDV graphs Given that the dually chordal graphs were deﬁned as the clique graphs of the chordal graphs, it is interesting to determine whether the application of the clique operator to some subclasses of the chordal graphs gives rise to new subclasses of the dually chordal graphs. This approach does not work in several cases. For example, K(DOUBLY CHORDAL) = DOUBLY CHORDAL [3], K(ST RONGLY CHORDAL) = ST RONGLY CHORDAL [1] and K(UV ) = CHORDAL [25]. Notwithstanding, new classes are obtained when we use DV and RDV graphs. A graph G is dually DV if it is the clique graph of a DV graph. It is dually RDV if it is the clique graph of an RDV graph. Like dually chordal graphs, both classes have characteristic trees.

Theorem 1.21. Let G be a connected graph. Then:

(a) G is dually DV if and only if there exists a directed spanning tree T of G such that every clique of G induces a directed path of T .

(b) G is dually RDV if and only if there exists a rooted directed spanning tree T of G such that every clique of G induces a directed path of T .

The tree described in (a) is called a DV compatible tree. The tree described in (b) is called an RDV compatible tree. Much like the suns can be used to establish the diﬀerence between chordal, DV and RDV graphs, the clique graphs of the suns are useful to diﬀerentiate the dual classes. The wheel Wn, n ≥ 4, is a graph such that V (Wn) = {u, v1, ..., vn}, where {v1, ..., vn} induces a chordless cycle and u is a universal vertex. It is easy to see that, for all k ≥ 4, the clique graph of the k-sun is Wk. It is possible to use Theorem 1.21 to verify that, for all odd k, Wk is dually chordal but not dually DV and, for all even k, Wk is dually DV but not dually RDV .

17 Chapter 2

Special eccentric vertices in chordal graphs, dually chordal graphs and their subclasses

In this chapter, we ﬁnd analogies between the linear orderings of chordal, doubly chordal, strongly chordal, power chordal and dually chordal graphs and between the vertices distin- guishing them. This will be mainly done through a study of eccentric vertices, as it will be shown in Theorems 2.3, 2.7, 2.9, 2.10 and 2.13. First of all, note that simplicial vertices and vertices with a maximum neighbor are generally dominated vertices. We will focus on metric aspects of the graphs, so the following property is of interest:

Proposition 2.1. Let G be a graph and u, v, w, x be vertices of G such that w is diﬀerent from the other three and w is dominated by x. Then, dG−w(u, v) = dG(u, v).

Proof. If there is no uv-path in G, then dG−w(u, v) = dG(u, v) = ∞. Otherwise, let P be a uv-path in G of minimum length. If w is not a vertex of P , then P is also a path in G − w and hence dG−w(u, v) = dG(u, v). If w is a vertex of P , let y1...yiwyi+1...yj be the sequence of vertices of P , with y1 = u and yj = v. Now we prove that x is not a vertex of P . As w is dominated by x, we have that yi, yi+1 ∈ N[x]. If there exists k such that k ≤ i and yk = x, then y1...ykyi+1...yj is a uv-path shorter than P , which is a contradiction. Therefore, for all 1 ≤ k ≤ i, yk 6= x. Similarly, for all i + 1 ≤ k ≤ j, yk 6= x. We infer that y1...yixyi+1...yj is a uv-path in G − w with the same length as P . Therefore, the equality dG−w(u, v) = dG(u, v) follows.

There is something else we can say about removing dominated vertices. Although not all the classes we have mentioned above are hereditary, we can see that all of them are closed under the removal of dominated vertices.

Proposition 2.2. Let G be a chordal/dually chordal/doubly chordal/power chordal/strongly chordal graph and v be a vertex of G dominated by another vertex w. Then, G−v is chordal/dually chordal/doubly chordal/power chordal/strongly chordal.

18 Proof. Since the classes of chordal graphs and strongly chordal graphs are hereditary, the proof is direct if G is chordal or if G is strongly chordal. Now we consider the remaining cases: 0 G is dually chordal: Let C be a clique of G − v. Deﬁne C = C ∪ {v} if C ⊆ NG[v] and C0 = C otherwise. It is simple to see that C0 is a clique of G. Now we prove that G − v is clique-Helly and K(G − v) is chordal. 0 0 0 Let C1,C2, ..., Cn be pairwise intersecting cliques of G − v. Then, C1,C2, ..., Cn are pairwise intersecting cliques of G. Since G is dually chordal, G is clique-Helly, so we can take a vertex 0 0 0 u ∈ C1 ∩ C2 ∩ ... ∩ Cn. If u 6= v, then u ∈ C1 ∩ C2 ∩ ... ∩ Cn. If u = v, then, for all 1 ≤ i ≤ n, 0 w ∈ Ci because v is dominated by w. Thus, w ∈ C1 ∩ C2 ∩ ... ∩ Cn. Therefore, G − v is clique-Helly. We deduce from the previous reasoning that, given C1,C2 ∈ C(G − v), C1 ∩ C2 6= ∅ if and 0 0 only if C1 ∩ C2 6= ∅. Therefore, K(G − v) is an induced subgraph of K(G). Since G is dually chordal, K(G) is chordal. Thus, K(G − v) is chordal. By Theorem 1.16, G − v is dually chordal. G is doubly chordal: we know from the above that G − v is chordal and dually chordal. Therefore, G − v is doubly chordal. G is power chordal: We know that G − v is chordal. Now we show that (G − v)2 = G2 − v. It is clear that every edge of (G − v)2 is an edge of G2 − v. The converse follows from 2.1. Consequently, (G − v)2 is an induced subgraph of G2, which is chordal. Thus, (G − v)2 is chordal. Therefore, G − v is power chordal.

Now we go back to chordal graphs. As we saw in Chapter 1, the existence of a perfect elimination ordering in a chordal graph is a direct consequence of Theorem 1.2. Theorem 1.2 is a classic theorem in the history of chordal graphs. But some properties which are more speciﬁc on how to locate the simplicial vertices were proved later. For example:

Theorem 2.3. [9] Let G be a chordal graph and v ∈ V (G). Then, v has an eccentric vertex which is simplicial.

Proof. A proof is included because of the importance of this theorem in the development of the chapter. The proof is trivial if G has only one vertex, so we assume that G has at least two vertices. We first prove that there exists a perfect elimination ordering for G such that v is the last vertex of it. If G is complete, then every order of the vertices makes a perfect elimination ordering, so we can pick one ending in v. If G is not complete, then there exist two nonadjacent simplicial vertices v1 and v2. Suppose without loss of generality that v 6= v1. Then, we make v1 be the first vertex of a perfect elimination ordering. In order to obtain the next vertex of the ordering, repeat the same reasoning on the graph G − v1. Continue removing the vertices obtained and applying the reasoning until a perfect elimination ordering is complete. Let v1...vn−1vn, vn = v, be the ordering obtained and let vj be the first eccentric vertex of v appearing in the ordering. Now we prove that vj is simplicial in G. Suppose that vj is not simplicial in G. Then, there exist two nonadjacent vertices x and y in N[vj]. Since vj is simplicial in Gj, x 6∈ V (Gj) or y 6∈ V (Gj). Suppose without loss of generality that x 6∈ V (Gj) and that x = vi, for some i < j. As vi is simplicial in Gi, the closed neighborhood of vi in Gi is contained in the closed neighborhood of vj in Gi. Thus, the distance

19 from vi to v in Gi is greater than or equal to the distance from vj to v in Gi. This allows us to conclude, applying Proposition 2.1 as many times as necessary, that the distance from vi to v in G is greater than or equal to the distance from vj to v in G. Consequently, since vj is an eccentric vertex of v in G, vi is eccentric as well, contradicting the way vj was chosen. Therefore, vj is simplicial.

Figure 2.1: A 4-sun, a vertex v of it and a simplicial vertex w which is an eccentric vertex of v.

It is interesting to see that Theorem 2.3 implies Theorem 1.2. Theorem 2.3 clearly implies the existence of a simplicial vertex. Let v be a simplicial vertex of G. If ecc(v) ≤ 1, then G is complete. If ecc(v) > 1, let w be a simplicial vertex of G which is eccentric to v. Then, v and w are two nonadjacent simplicial vertices of G. Theorem 2.3 is the cornerstone of this chapter. It is our interest to find out whether we can find an analogous property about dually chordal graphs. More specifically, given a dually chordal graph G and v ∈ V (G), we wonder whether v has an eccentric vertex with a maximum neighbor. This is found to be true and not complicated to prove. As a result, we get additional properties that remind us of others appearing in the literature on chordal graphs. For this, we need some previous results. Lemma 2.4. [3] Let G be a dually chordal graph and A ⊆ V (G) be such that d(x, y) ≤ 2 for all x, y ∈ A. Then, there exists a vertex w such that A ⊆ N[w]. Lemma 2.5. [3] Let G be a dually chordal graph. Then, G2 is chordal. Lemma 2.6. Let G be a dually chordal graph and v be a simplicial vertex of G2. Then, v has a maximum neighbor in G.

2 2 Proof. As v is simplicial in G , the distance in G between every pair of vertices of NG[v] is at 2 most 2. Then, by Lemma 2.4, there exists a vertex w such that NG[v] ⊆ NG[w]. Therefore, w is a maximum neighbor of v.

The ﬁrst major result can now be proved.

20 Theorem 2.7. Let G be a dually chordal graph and v be a vertex of G. Then, there exists an eccentric vertex of v with a maximum neighbor.

0 2 Proof. Suppose first that eccG(v) is odd and let v be an eccentric vertex of v in G. As G is chordal, Theorem 2.3 implies that there exists a vertex w which is simplicial in G2 and is eccentric to v in G2. Hence, by Lemma 2.6, w has a maximum neighbor in G. We prove now that w is also eccentric to v in G. Note first that, by the definition of G2, if two vertices are at distance k in G, then their 2 k k+1 0 eccG(v)+1 distance in G is 2 if k is even or 2 if k is odd. Then, as eccG(v) is odd, dG2 (v, v ) = 2 . Furthermore, every vertex eccentric to v in G is also eccentric to v in G2. Thus, v0 is eccentric 2 2 eccG(v)+1 to v in G and hence the eccentricity of v in G equals 2 . Since w is also eccentric to v 2 eccG(v)+1 in G , we infer that dG2 (v, w) = 2 . 2 eccG(v)+1 By using the definition of G again and that dG2 (v, w) = 2 , we obtain two possible values for dG(v, w), namely, eccG(v) and eccG(v) + 1. Thus, dG(v, w) ≥ eccG(v). Furthermore, the definition of eccentricity implies that dG(v, w) ≤ eccG(v). Therefore, dG(v, w) = eccG(v) and hence w is the required vertex. 0 ∗ If eccG(v) is even, let G be the graph obtained from G by adding a new vertex v and making 0 it adjacent to v. Then, G is dually chordal. In fact, if v1...vn is a maximum neighborhood ∗ 0 ordering for G, then v v1...vn is a maximum neighborhood ordering of G . As eccG(v) is even, ∗ eccG0 (v ) is odd and, by the previous case, there exists a vertex u with a maximum neighbor in 0 ∗ ∗ G such that dG0 (v , u) = eccG0 (v ). It is clear that u is eccentric to v in G. Now we show that u has a maximum neighbor in G. Let w be a maximum neighbor of u in G0. If w = v∗, then u = v. As v∗ is a maximum neighbor of v, v∗ dominates v. By the construction of G0, v dominates v∗ as well. Thus, ∗ ∗ NG0 [v] = NG0 [v ] = {v, v }. We infer that v is adjacent to no other vertex of G and hence is a maximum neighbor of itself in G. ∗ 2 2 If w 6= v , then w ∈ V (G). The inclusion NG0 [u] ⊆ NG0 [w] implies that NG[u] ⊆ NG[w]. Therefore, w is a maximum neighbor of u in G. If eccG(v) = ∞, then G is not connected. Let w be a vertex with a maximum neighbor in a connected component of G different from that of v. Thus, w is the desired vertex.

Much like Theorem 2.3 can be used to prove basic properties of chordal graphs, now we will see that Theorem 2.7 is equally useful and that it allows to ﬁnd connections between the approaches whereby chordal graphs and dually chordal graphs can be studied.

Corollary 2.8. Let G be a dually chordal graph with more than one vertex. Then,

(a) G is complete or it has two nonadjacent vertices each with a maximum neighbor.

(b) There exist two vertices v1 and v2 each with a maximum neighbor and such that d(v1, v2) = diam(G).

Proof. (a): Let v be a vertex of G with a maximum neighbor. If ecc(v) = 1, then v is a universal vertex and every vertex of G has v as a maximum neighbor. Thus, G is complete or there are two nonadjacent vertices having v as a maximum neighbor. If ecc(v) > 1, let w be an eccentric vertex of v with a maximum neighbor. Then, v and w are two nonadjacent vertices each with a maximum neighbor.

21 (b): Let k = diam(G) and x, y be two vertices such that d(x, y) = k. Then, there exists a vertex v1 with a maximum neighbor which is eccentric to x, so d(x, v1) = k. Likewise, there exists a vertex v2 with a maximum neighbor which is eccentric to v1. Consequently, d(v1, v2) = k. For the remainder of the chapter, the same type of study is performed on power chordal, doubly chordal and strongly chordal graphs. Given the clarity of the goals, no further discussion of the subject appears. The focus will be on stating the theorems and giving their proofs. Theorem 2.9. Let G be a power chordal graph and v ∈ V (G). Then, v has an eccentric vertex which is power simplicial.

Proof. As a consequence of Theorem 1.9, there exists a power perfect ordering v1...vn of G such that vn = v. Suppose first that eccG(v) is odd. Let vj be the first eccentric vertex of v appearing in the ordering. As v1...vn is a perfect elimination ordering of G, we can repeat the arguments of the proof of Theorem 2.3 to conclude that vj is simplicial in G. 2 By construction, v1...vn is also a perfect elimination ordering of G . Since eccG(v) is odd, 2 vj is the first eccentric vertex of v in G appearing in the ordering. Apply the same arguments 2 again to conclude that vj is simplicial in G . Therefore, vj is a power simplicial vertex of G. 0 0 If eccG(v) is even, construct the graph G as in the proof of Theorem 2.7. Then, G is power ∗ 0 ∗ chordal and v v1...vn is a power perfect elimination ordering of G . In addition, eccG0 (v ) is odd. Use the first part of this proof to obtain a vertex w which is power simplicial and eccentric to v∗ in G0. By Proposition 2.1, the distances between vertices of G are the same in G and G0. We can conclude from this that w is also power simplicial in G. Finally, w is clearly eccentric to v in G. If eccG(v) = ∞, take a power simplicial vertex in a connected component of G different from that of v and it will clearly be eccentric to v.

Theorem 2.10. Let G be a doubly chordal graph and v ∈ V (G). Then, there exists an eccentric vertex of v which is doubly simplicial. Proof. As G is dually chordal, we can use Lemma 2.5 to infer that G2 is chordal. Thus, G is power chordal. By Theorem 2.9, there exists a vertex w which is simplicial in both G and G2 and is eccentric to v in G. By Lemma 2.6, w has a maximum neighbor in G, so it is doubly simplicial.

Corollary 2.11. Let G be a power/doubly chordal graph with more than one vertex. Then,

(a) G is complete or there exist two nonadjacent power/doubly simplicial vertices.

(b) There exist two power/doubly simplicial vertices v1 and v2 such that d(v1, v2) = diam(G).

The proofs are omitted because they can be obtained through reasonings that are almost identical to some used before. It only remains to consider the class of strongly chordal graphs. Lemma 2.12. Let G be a graph and u, v, w be three vertices of G such that w is a maximum neighbor of v, ecc(w) > 1 and 2 ≤ d(u, v) < ∞. Then, d(u, v) = d(u, w) + 1 and the set of eccentric vertices of w is equal to the set of eccentric vertices of v.

22 Proof. If d(u, v) = 2, then d(u, w) = 1 because of the deﬁnition of maximum neighbor. Suppose now that d(u, v) > 2. By the triangle inequality, d(u, v) ≤ d(u, w) + d(w, v), that is, d(u, v) ≤ d(u, w) + 1. Let vv1v2...u be a shortest vu-path. Then, wv2...u is a wu-path of length d(u, v) − 1. Thus, d(u, v)−1 ≥ d(u, w) and hence d(u, v) ≥ d(u, w)+1. Therefore, the equality d(u, v) = d(u, w)+1 holds. Combine this with the inequality ecc(w) > 1 to deduce that every vertex eccentric to v is at a distance greater than or equal to 3 from v and, consequently,

d(v, u) = ecc(v) ⇔ d(v, u) = max{d(v, x): x ∈ V (G)} ⇔ d(v, u) = max{d(v, x): x ∈ V (G), d(v, x) ≥ 3} ⇔ d(w, u) + 1 = max{d(w, x) + 1 : x ∈ V (G), d(w, x) ≥ 2} ⇔ d(w, u) = max{d(w, x): x ∈ V (G), d(w, x) ≥ 2} ⇔ d(w, u) = max{d(w, x): x ∈ V (G)} ⇔ d(w, u) = ecc(w)

Theorem 2.13. Let G be a strongly chordal graph and v ∈ V (G). Then, there exists an eccentric vertex of v that is simple.

Proof. The proof will be by induction on n = |V (G)|. The statement of the theorem is obviously valid when n = 1. Suppose now that it is true when n = k, where k is a number greater than or equal to 1, and that G is a strongly chordal graph with k + 1 vertices. We assume that G is connected; otherwise, a simple eccentric vertex can be trivially found. The proof will be divided into cases. Case 1: G has a universal vertex. Let w be a universal vertex of G. If w is simple, then G is complete because simple vertices are simplicial. Thus, the existence of an eccentric simple vertex is evident. If w is not simple and v = w, then the fact that w is universal implies that any simple vertex of G is an eccentric vertex of v. Now assume that v 6= w and that w is not simple. Consider the strongly chordal graph G − w. In case that G − w is not connected, any simple vertex in G − w located in a connected component different from that of v is an eccentric simple vertex of v in G. If G−w is connected, applying the inductive hypothesis to it yields an eccentric simple vertex u for v in G − w. It is not difficult to see that u is simple and eccentric to v in G as well. Case 2: G has no universal vertices. Case 2.a: v is simple. As v is simple, we can take a vertex v0 that is a maximum neighbor of v. Then, v 6= v0 because v is not universal. By the inductive hypothesis, there exists a vertex w that is simple and eccentric to v0 in G−v. Note that d(v0, w) ≥ 2 because otherwise v0 would be universal in G. As v0 is a maximum neighbor of v in G, and hence N 2[v] ⊆ N[v0], we conclude that d(v, w) ≥ 3. Thus, the closed neighborhoods of the vertices in N[w] are equal in G and G − v, which implies that w is also simple in G. By Lemma 2.12, w is eccentric to v in G as well. Case 2.b: v is not simple and there is a simple vertex that is not adjacent to v. Let w be a simple vertex not adjacent to v. If it is eccentric to v, then we are done. If not, consider the strongly chordal graph G − w which, by inductive hypothesis, possesses a simple vertex w0 that is eccentric to v. Then, as a consequence of Proposition 2.1, w0 is also eccentric to v in G, so it suffices to prove that w0 is simple in G.

23 Suppose on the contrary that w0 is not simple in G. We ﬁrst prove that w ∈ N 2[w0]. As w0 0 is not simple, there are vertices u1 and u2 in N[w ] such that N[u1] * N[u2] and N[u2] * N[u1]. 0 2 0 If u1 = w or u2 = w, then w is in N[w ], and so is in N [w ]. 0 If u1 6= w and u2 6= w, then u1 and u2 are in the closed neighborhood of w in G − w. Since 0 w is simple in G − w, we may assume without loss of generality that NG−w[u1] ⊆ NG−w[u2]. If w 6∈ NG[u1], then NG[u1] ⊆ NG[u2], thus contradicting our previous assumption. Therefore, 0 2 0 w ∈ NG[u1] which, added to the fact that u1 ∈ NG[w ], implies that w ∈ N [w ]. 0 0 Let u be a maximum neighbor of w in G. Then, u ∈ NG[w ] and hence d(v, w ) ≤ d(v, u) + 1. Combine this with Lemma 2.12 to obtain that d(v, w0) ≤ d(v, w), contradicting that w is not an eccentric vertex of v in G. Therefore, w0 is necessarily simple in G. Case 2.c: v is not simple and v is adjacent to all the simple vertices of G. We prove that diam(G) ≤ 2. Suppose on the contrary that diam(G) ≥ 3. Let x and y be vertices such that d(x, y) = diam(G). Thus, {x, y} * N[v], so we can assume without loss of generality that x 6∈ N[v]. Since all the simple vertices are simplicial and adjacent to v, we conclude that none of them is adjacent to x. Then, by case 2.b, x has a simple eccentric vertex x0 and hence d(x, x0) = diam(G). By case 2.a, we know that x0 has a simple eccentric vertex x00, so d(x0, x00) = diam(G). Since both x0 and x00 are adjacent to v, we conclude that d(x0, x00) ≤ 2, which is a contradiction. Therefore, diam(G) ≤ 2. Since G is dually chordal, we can apply Lemma 2.4 to conclude that G has a universal vertex, contradicting the initial assumption of case 2. Therefore, case 2.c is not possible. As all the possible cases have been considered, the proof is now complete.

Figure 2.2: A strongly chordal graph, a vertex v and a simple vertex w which is eccentric to v.

Corollary 2.14. Let G be a strongly chordal graph with more than one vertex. Then,

(a) G is complete or it has two nonadjacent simple vertices.

(b) There exist two simple vertices v1 and v2 such that d(v1, v2) = diam(G).

24 Chapter 3

Minimal vertex separators of dually chordal graphs

We have seen before that many characterizations of the dually chordal graphs are known, mainly involving cliques and neighborhoods. However, not much had been revealed about their minimal vertex separators. For that reason, one of the purposes of this chapter is to study minimal vertex separators of dually chordal graphs to determine if the properties known about cliques and neighborhoods have their counterparts when dealing with minimal vertex separators. The chapter has four sections. In the first section, we study the relationship between compatible trees and minimal vertex separators. In the second section, we find properties of the minimal vertex separators of dually chordal graphs that are contained in closed neighborhoods. In the third section, we see how what was proved in the first section can be used to characterize dually chordal graphs. Some of the characterizations are used in the last section to obtain a new proof about hereditary dually chordal graphs, i.e., about strongly chordal graphs. All the graphs considered will be assumed to be connected for a better handling of the proofs of the properties we are going to discuss. It is not difficult to extend them to disconnected graphs by applying the proofs to each connected component.

3.1 Minimal vertex separators and compatible trees

We have seen that compatible trees have many interesting properties, which allow to explain why they can be so helpful in modeling dually chordal graphs. The more we know about them, the more evident their importance is. In this section, we show that they are related to minimal vertex separators in a way very similar to the known relationships with cliques and neighborhoods. More precisely, the main goal is to show that every minimal vertex separator induces a subtree of every compatible tree.

Theorem 3.1. Let G be a dually chordal graph, T be a tree compatible with G, u, v be two nonadjacent vertices of G and S be a minimal uv-separator. Then, S induces a subtree of T .

Proof. If S consists of only one vertex, then the proof is trivial. Otherwise, we have to prove that every pair of distinct vertices x and y of S satisfy that T [x, y] ⊆ S. If T [x, y] = {x, y}, then the inclusion is clear. Now consider the case that T [x, y] 6= {x, y}. Assume that T [x, y] is not a subset of S and let z be a vertex in T (x, y) \ S. Let also x0 be the last vertex preceding z in T hx, yi such that x0 ∈ S and y0 be the ﬁrst vertex following z in T hx, yi such that y0 ∈ S. Thus, T (x0, y0) ∩ S = ∅.

25 We continue by proving a series of facts: 0 0 • u 6∈ T [x , v]: Suppose on the contrary that u ∈ T [x , v]. Let v1v2...vn, v1 = u, vn = v, be a 0 uv-path such that its only vertex in S is x and let i = max {j : u ∈ T [vj, v]}. Now consider the graph T − u. If vi and vi+1 are in the same connected component of T − u, then u ∈ T [vi+1, v] (see Figure 3.1), contradicting the deﬁnition of i. Thus, vi and vi+1 are in diﬀerent connected components of T − u and hence u ∈ T [vi, vi+1].

Figure 3.1: If u ∈ T [vi, v] and vi and vi+1 are in the same connected component of T − u, then u ∈ T [vi+1, v].

We can apply Theorem 1.15 to conclude that u is adjacent to vi+1. Then, uvi+1...v is a uv-path which, by construction, does not contain x0 as a vertex and hence contains no vertex of S, contradicting the assumption that S is a uv-separator. Therefore, u 6∈ T [x0, v]. • Through a similar reasoning, we can prove that u 6∈ T [y0, v], v 6∈ T [u, x0] and v 6∈ T [u, y0]. • T (x0, y0) is contained in the same connected component of G − S as u: Let G[A] be the connected component of G − S containing u and G[B] be the connected component of G − S containing T (x0, y0). Consider the graph T − T (x0, y0). If A intersects more than one connected component of T − T (x0, y0), then we can take two vertices v1, v2 ∈ A such that v1 is adjacent to v2 and they are in diﬀerent connected components 0 0 0 0 of T − T (x , y ). Then, T (v1, v2) ∩ T (x , y ) 6= ∅. Let w be a vertex of that intersection. Then, by Theorem 1.15, w is adjacent to v1 and v2, so w ∈ A. Therefore, A ∩ B 6= ∅, implying that A = B. Suppose now that A intersects only one connected component of T − T (x0, y0). Let D be the set of vertices of that component. Since x0 and y0 are in diﬀerent connected components of T − T (x0, y0), x0 6∈ D or y0 6∈ D. Assume without loss of generality that y0 6∈ D. As S is a minimal uv-separator and y0 ∈ S, y0 is adjacent to at least one vertex w ∈ A. If w ∈ T (x0, y0), then A ∩ B 6= ∅ and hence A = B. If w 6∈ T (x0, y0), then w ∈ D which, combined with the fact that y0 6∈ D, implies that there exists a vertex w0 ∈ T (w, y0) ∩ T (x0, y0). Since T is a compatible tree, w is adjacent to w0 and hence these two vertices are in the same connected component of G − S. Thus, A ∩ B 6= ∅, so A = B. • We can prove with a similar argument that T (x0, y0) is contained in the same connected component of G − S as v. The last two facts imply that u and v are in the same connected component of G − S, contradicting the fact that u and v are separated by S. Therefore, T [x, y] ⊆ S. This is what we needed to complete the proof.

26 An immediate consequence of the previous theorem is that every minimal vertex separator of a dually chordal graph induces a connected subgraph. There is more that we can conclude. Proposition 3.2. [12] The family of all subtrees of a tree is Helly. Then, the following corollary follows: Corollary 3.3. Let G be a dually chordal graph. Then, S(G) is Helly, its intersection graph is chordal and every member of S(G) induces a connected subgraph of G. It will be proved in the third section of this chapter that the converse is also true.

3.2 Minimal vertex separators and neighborhoods

This section deals with three questions, namely, how and where minimal vertex separators contained in the closed neighborhood of a vertex can be found; and what the necessary and suﬃcient conditions for every minimal vertex separator of a dually chordal graph to be contained in the closed neighborhood of a vertex are. These conditions are given in Theorem 3.6. To start, we note that the characterization of chordal graphs in Theorem 1.1 implies that, given two nonadjacent vertices u and v, there exists another vertex w whose closed neighborhood (excluding u or v if any of them is a neighbor of w) separates u and v. In the following, we show that the same is true for dually chordal graphs. Theorem 3.4. Let u and v be two nonadjacent vertices of a dually chordal graph G. Then, there exists a vertex w, w 6= u and w 6= v, such that N[w] \{u, v} is a uv-separator. Proof. Let T be a tree compatible with G. Note that u and v are not adjacent in T because they are not adjacent in G, so T (u, v) is not empty. Let w be any vertex of T (u, v). If P is a uv-path in G, then it has vertices in diﬀerent connected components of T − w. This implies that P has two consecutive vertices x1 and x2 such that w ∈ T [x1, x2]. Since T is compatible with G, {x1, x2} ⊆ N[w]. As u and v are not adjacent, unlike x1 and x2, x1 or x2 belongs to N[w] \{u, v}. Consequently, every uv-path in G has some vertex in N[w] \{u, v}, that is, N[w] \{u, v} is a uv-separator of G.

Every vertex separator obviously contains a minimal vertex separator, which combined with the previous proof gives the following corollary: Corollary 3.5. Let G be a dually chordal graph, T be a tree compatible with G and u and v be two nonadjacent vertices of G. Then, the closed neighborhood of every vertex in T (u, v) contains a minimal uv-separator. Despite this corollary, not every minimal vertex separator of a dually chordal graph is necessarily contained in a closed neighborhood (see Figure 3.2). Notwithstanding, this property becomes true under additional conditions. In fact, we get a necessary and suﬃcient condition for every minimal vertex separator of a dually chordal graph to be contained in a closed neighborhood. Theorem 3.6. Let G be a dually chordal graph. Then, every minimal vertex separator of G is contained in a closed neighborhood if and only if every chordless cycle of length at least four is contained in a closed neighborhood.

27 Figure 3.2: A dually chordal graph with the edges of a compatible tree in color. {1, 4, 7, 8} is a minimal 26-separator, but it is not contained in the closed neighborhood of any vertex.

Proof. Suppose that G is a dually chordal graph with every minimal vertex separator contained in a closed neighborhood. Let C be a chordless cycle of G of length at least four, if any, and let x, y be two nonconsecutive vertices of C. As there are two xy-paths in C of length at least two, we can take inner vertices u and v of each of those paths. For each uv-path in G not completely contained in G[C], pick one vertex z 6∈ C. Let R be the collection of the vertices chosen. Set S = R ∪ {x, y}. Now we prove that S is a uv-separator. We just need to prove that every uv-path P has a vertex in S. If P is not a path in G[C], then it has a vertex in S because of the way R was constructed. Moreover, since C is chordless, there are only two uv-paths in G[C], with x and y in each of them. 0 Let S be a minimal uv-separator contained in S and let P1 and P2 be the uv-paths in G[C] containing x and y, respectively. As x is the only vertex in S ∩ V (P1) and y is the only vertex 0 0 in S ∩ V (P2), x and y are in S . By our assumption, S is contained in the closed neighborhood of a vertex and hence N[x] ∩ N[y] 6= ∅. Therefore, d(x, y) = 2. We conclude that every two vertices of C are either adjacent or are at distance two. By Lemma 2.4, there exists a vertex w such that C ⊆ N[w]. Conversely, suppose that every chordless cycle whose length is greater than or equal to four is contained in the closed neighborhood of a vertex and let S be a minimal separator of two nonadjacent vertices a and b. The proof that S is contained in a closed neighborhood can be done by adapting the proof that a graph is chordal if and only if every minimal vertex separator is complete, as given in [12]. Let x and y be two vertices in S. Consider a cycle C containing x and y constructed as in [12]. More speciﬁcally, take two paths xa1...ary and yb1...btx where, for all 1 ≤ i ≤ r, ai is in the same connected component of G − S as a and, for all 1 ≤ i ≤ t, bi is in the same connected component of G − S as b, such that these paths are chosen to be of smallest possible length. Thus, C is the cycle xa1...aryb1...btx. If C has a chord, then xy is the only possible chord, so x and y are adjacent. If C is chordless, then it is contained in the closed neighborhood of a vertex w and hence xwy is a xy-path of length two. Therefore, for all x, y ∈ S, d(x, y) ≤ 2. By Lemma 2.4, S is contained in the closed neighborhood of a vertex.

28 If C is a cycle of length four or ﬁve, then every two vertices of it are at distance not greater than two and hence, by Lemma 2.4, C is contained in the closed neighborhood of a vertex. We obtain the following corollary: Corollary 3.7. Let G be a dually chordal graph such that G is chordal or every chordless cycle of G has length at most ﬁve. Then, every minimal vertex separator of G is contained in the closed neighborhood of a vertex. It is clear from this that every dually chordal graph with at least one minimal vertex separator not contained in a closed neighborhood must have a chordless cycle of length at least six not contained in a closed neighborhood. In Figure 3.2, that cycle is induced by {1, 2, 3, 4, 5, 6}.

We will study in the remainder of this section a particular case of minimal vertex separators of dually chordal graphs contained in a closed neighborhood, i.e., those whose vertices are equidistant from another vertex. A link with the Helly property will be found to be very useful. Thus far, three Helly families of sets of vertices of a dually chordal graph were identiﬁed, namely, the family of its cliques, the family of its closed neighborhoods (and its disks) and the family of its minimal vertex separators. The fact that every member of these families induces a subtree of a ﬁxed compatible tree implies, by Proposition 3.2, that the union of the three families is itself Helly. Proposition 3.8. Let G be a dually chordal graph. Then, C(G)∪S(G)∪D(G) is a Helly family. This proposition has implications for some of the issues discussed before. When we considered a minimal vertex separator S contained in the closed neighborhood of a vertex w, it could happen that w 6∈ S. Now we can prove that, in dually chordal graphs, w can always be chosen in such a way that it is an element of S. Proposition 3.9. Let S be a minimal vertex separator of a dually chordal graph such that S is contained in the closed neighborhood of a vertex. Then, there exists a vertex w such that w ∈ S and S ⊆ N[w]. Proof. Consider the family composed of S and the closed neighborhoods of the vertices in S. Since S is contained in the closed neighborhood of a vertex, this family is intersecting. Then, by Proposition 3.8, there exists a vertex w which is an element of every member of the family, that is, w ∈ S and w is in the closed neighborhood of every vertex of S. The latter implies that S ⊆ N[w].

For a vertex v of a graph G and a nonnegative integer i, deﬁne D(v, i) = {w ∈ V (G): d(v, w) = i}. The following result is true for all graphs: Lemma 3.10. Let u and v be two nonadjacent vertices of a graph G such that d(u, v) = k. Then, for every 1 ≤ i < k, there exists a minimal uv-separator Si contained in D(v, i). Proof. We prove that, for all 1 ≤ i ≤ k − 1, D(v, i) is a uv-separator and hence it contains a minimal uv-separator. We need to show that every uv-path v1v2...vn, v1 = u, vn = v, has a vertex in D(v, i). Let m = min {j : d(vj, v) ≤ i}. Then, m ≥ 2. If d(vm, v) < i, then, by the triangle inequality, d(vm−1, v) ≤ d(vm−1, vm) + d(vm, v) < 1 + i. Hence, d(vm−1, v) ≤ i, contradicting the deﬁnition of m. Therefore, d(vm, v) = i, i.e., vm ∈ D(v, i). In the particular case of dually chordal graphs, we can see that a minimal vertex separator like that is contained in a closed neighborhood.

29 Proposition 3.11. Let G be a dually chordal graph and V 0 be a subset of V (G) such that V 0 is contained in the closed neighborhood of some vertex w. If all the vertices of V 0 are at distance 0 i from a certain vertex v 6∈ V , then there exists a vertex vi such that d(v, vi) = i − 1 and 0 V ⊆ N[vi]. Proof. Consider the family composed of N i−1[v] and the closed neighborhood of each vertex of 0 0 V . Since V ⊆ N[w], this family is intersecting. By Proposition 3.8, there exists a vertex vi 0 that is in every member of the family. Therefore, V ⊆ N[vi] and d(v, vi) = i − 1. Proposition 3.12. Let G be a dually chordal graph, T be a tree compatible with G, T 0 be a subtree of T and v ∈ V (G) \ V (T 0). If all the vertices of T 0 are at a distance i from v, then there 0 exists a vertex vi such that d(v, vi) = i − 1 and V (T ) ⊆ N[vi]. Proof. Consider the family {T [v, w]: w ∈ V (T 0)} ∪ {V (T 0)}. This family is intersecting. Then, by Proposition 3.2, we can pick a vertex x which is in all the members of the family. Now we show that V (T 0) ⊆ N[x]. If V (T 0) = {x}, then clearly V (T 0) ⊆ N[x]. Otherwise, let w ∈ V (T 0), w 6= x, and consider a vw-path P in G of length i. Let y be the vertex preceding w in P , so d(v, y) = i − 1. We can see that w and y are in diﬀerent connected components of T − x. Otherwise, since x ∈ T [v, w], also x ∈ T [v, y]. But this contradicts the fact that v, y ∈ N i−1[v] and that N i−1[v] induces a subtree of T . We infer that x ∈ T [w, y]. As T is compatible with G, w is adjacent to x. This way, the inclusion V (T 0) ⊆ N[x] follows. We can now apply Proposition 3.11 to derive 0 the existence of a vertex vi such that V (T ) ⊆ N[vi] and d(v, vi) = i − 1. Corollary 3.13. Let G be a dually chordal graph and u and v be two nonadjacent vertices such that d(u, v) = k. If Si is a minimal uv-separator contained in D(v, i), for some 1 ≤ i < k, then there exists a vertex vi such that d(v, vi) = i − 1 and Si ⊆ N[vi]. Proof. Since G is dually chordal, we can take a tree T compatible with G. By Theorem 3.1, T [Si] is a subtree of T . Then, Proposition 3.12 applied to T [Si] gives the desired result. As a consequence, we can prove the following interesting property:

Proposition 3.14. Let G be a dually chordal graph and u, v be two nonadjacent vertices of G such that d(u, v) = k. Then, there exists a path v1...vku, v = v1, and k − 1 disjoint minimal uv-separators S1, ..., Sk−1 such that, for all 1 ≤ i ≤ k − 1, Si ⊆ N[vi].

Proof. For each i, 1 ≤ i ≤ k − 1, let Si be a minimal uv-separator contained in D(v, i). We can apply Corollary 3.13 to get a vertex vi such that Si ⊆ N[vi] and d(v, vi) = i − 1. Let vk be any vertex adjacent to u and at distance k − 1 from v. We shall prove that v1...vku is a path satisfying the required conditions. By the construction, d(v, v1) = 0, so v1 = v. Therefore, it remains to prove that, for all 1 ≤ i ≤ k − 1, vi and vi+1 are adjacent. We prove it ﬁrst for 1 ≤ i ≤ k − 2. Let Q be a shortest uv-path and let z be the vertex of Q at distance i + 1 from v and at distance k − i − 1 from u. As z is the only vertex of Q at distance i + 1 from v, we conclude that z is necessarily an element of Si+1. Then, vi+1 is adjacent to z and hence, by the triangle inequality, d(vi+1, u) ≤ d(vi+1, z) + d(z, u) ≤ k − i. Also recall that d(vi+1, v) = i, so vi+1 can be found in a shortest uv-path. Consequently, vi+1 is necessarily in Si. As vi is adjacent to all the vertices of Si, vi is adjacent to vi+1. For i = k − 1, it is an immediate consequence of our choice that vk ∈ Sk−1. Therefore, vk−1 is adjacent to vk. This concludes the proof.

30 3.3 Characterizations

Recall that we have already proved that each minimal vertex separator of a dually chordal graph G induces a subtree of every compatible tree of G and hence induces a connected subgraph. We also saw that the family of minimal vertex separators is Helly and that its intersection graph is chordal. When we consider cliques, we ﬁnd properties that can be stated in similar terms: every clique induces a subtree of every compatible tree, on one hand; and C(G) is Helly and K(G) is chordal, on the other. Given that these two properties can be used to characterize dually chordal graphs, it is interesting to wonder if the same can be done if we use the properties of minimal vertex separators instead. Fortunately, we can answer in the aﬃrmative and this section is devoted to proving the characterizations of dually chordal graphs that can be obtained.

Theorem 3.15. A graph G is dually chordal if and only if it has a spanning tree T such that every minimal vertex separator of G induces a subtree of T .

Proof. By Theorem 3.1, it remains to prove only the backward direction. Assume that T is a spanning tree like the above described. Let x and y be two adjacent vertices of G and let z ∈ T (x, y). Suppose that x and z are not adjacent. Let S be a minimal xz-separator. As S must induce a subtree of T and must contain a vertex in T [x, z], T hy, zi is a yz-path in G − S (see Figure 3.3). We conclude that y and z are in the same connected component of G − S. But x is also in that component because it is adjacent to y, contradicting our assumption that x and z are separated by S. Thus, x and z are necessarily adjacent. Similarly, we can prove that y is adjacent to z. Therefore, by Theorem 1.15, G is dually chordal.

Figure 3.3: Since S induces a subtree and contains a vertex in T [x, z] (suppose that it is the blue one), the vertices of S are contained inside the circle. Therefore, T hy, zi is a yz-path in G − S.

If we analyze the steps of this proof, then it becomes clear that we might not need to know that every minimal vertex separator of G induces a subtree of T to conclude that G is dually chordal. Actually, the proof requires that, for every two nonadjacent vertices u and v, there exists at least one minimal uv-separator inducing a subtree of T . This leads to a slightly diﬀerent characterization of dually chordal graphs.

Theorem 3.16. Let G be a graph and F be a family of minimal vertex separators of G such that F contains a uv-separator for every two nonadjacent vertices u and v,. Then, G is dually chordal if and only if it has a spanning tree T such that every member of F induces a subtree of T .

31 The proof of Theorem 3.16 is essentially that of Theorem 3.15. The only diﬀerence is that we must choose S to be in F.

Now we present the second major characterization:

Theorem 3.17. A graph G is dually chordal if and only if every of its minimal vertex separators induces a connected subgraph, S(G) is Helly and the intersection graph of S(G) is chordal.

Proof. By Corollary 3.3, it remains to prove only the backward direction. Assume that every minimal vertex separator induces a connected subgraph, that the family of minimal vertex separators is Helly and that its intersection graph is chordal. The last two conditions imply that there exists a tree T whose vertex set is V (G) with each minimal vertex P separator inducing a subtree of T [23]. Choose T so that s(T ) := dG(v, w) is minimum. vw∈E(T ) If T is not a spanning tree of G, let v and w be two vertices adjacent in T but not in G, with d(v, w) = k. Consider the family F formed by all the minimal vertex separators containing both v and w, if there are any, one minimal vw-separator S1 contained in N[v] and one minimal vw-separator S2 contained in D(w, k − 1), whose existence is ensured by Lemma 3.10. We now prove that F is intersecting. Let S be a minimal vertex separator containing v and w. As S induces a connected subgraph by our assumption, there exists a vw-path P whose vertices are all in S. Since Si, i = 1, 2, separates v and w, it must contain a vertex of P . Hence, S ∩ Si 6= ∅. Let P be now a shortest vw-path. Then, P has a vertex x such that d(v, x) = 1 and d(w, x) = k − 1, which is unique in this respect. Consequently, x ∈ S1 ∩ S2. Thus, F is indeed an intersecting family. As S(G) is Helly, there exists a vertex u belonging to every member of F. Let T [A] and T [B] be the connected components of T − vw containing v and w, respectively. If u ∈ A, let T 0 = T − vw + uw. Now we show that every minimal vertex separator of G induces a subtree of T 0 (see Figure 3.4). In order to prove this, let S be a minimal vertex separator of G. If S ⊆ A or S ⊆ B, then S induces the same subtree in T and T 0. Otherwise, we have two vertices y, z ∈ S such that y ∈ A and z ∈ B. As S induces a subtree of T and v, w ∈ T [y, z], we conclude that v, w ∈ S and hence u ∈ S as well. Then, v and w are connected in T 0 by the path formed by merging T hv, ui = T 0hv, ui and uw. The vertices of this path are contained in S because u, v ∈ S and T [S] is connected. Moreover, the vertices of every other pair of elements of S that are adjacent in T are also adjacent in T 0. This is enough to conclude that S induces a subtree of T 0. However, s(T 0) < s(T ) because d(u, w) = k − 1 and d(v, w) = k, contradicting our choice of T . If u ∈ B, then we can remove vw and add uv to T and a similar contradiction arises. Thus, T is necessarily a spanning tree of G and hence, by Theorem 3.15, G is dually chordal.

Note that this characterization does not lead to an eﬃcient way to decide whether a graph is dually chordal or not. In order to test if the three conditions of the characterization are satisﬁed, we need a list of all the minimal vertex separators of the graph. However, there could be an exponential number of minimal vertex separators in a dually chordal graph. Consider, for example, the family of graphs shown in Figure 3.5. The graph Hn consists of n paths of length two, joining two vertices u and v, that do not share inner vertices. As every minimal uv-separator of Hn must contain one vertex from each path, we conclude that Hn has n 2 minimal uv-separators. If we add a new vertex to Hn and we make it universal, then the resulting graph is dually chordal and also has 2n uv-separators.

32 Figure 3.4: If every minimal vertex separator induces a subtree of T , then every minimal vertex separator induces a subtree of T 0 as well. A graphical idea of this fact can be found here for the cases that the separator is contained in A or in B (top) or that it has vertices of both sets (bottom).

Notwithstanding, we can get a condition characterizing dually chordal graphs depending on a family of minimal vertex separators with no more than 2|E(G)| members. In view of Theorem 3.16 and the steps of the proof of Theorem 3.17, the following can be proved:

Theorem 3.18. Let G be a graph and F be a family of minimal vertex separators of G such that, for every two nonadjacent vertices v and w of G, there exist minimal vw-separators S1 and k−1 k−1 S2 in F (they could be equal) such that S1 ∩ S2 6= ∅ and S1 ∩ S2 ⊆ N [v] ∩ N [w], where k is the distance from v to w in G. Then, G is dually chordal if and only if each member of F induces a connected subgraph of G, F is Helly and the intersection graph of F is chordal.

Observe that, in order to build a family like F, one of the possible ways to construct S1 and S2 is as in the proof of Theorem 3.17. Theorem 3.16 allows us to consider just the members of F instead of all the minimal vertex separators (if there were more). The condition that S1 ∩ S2 6= ∅ ensures that F deﬁned in a way similar to that of Theorem 3.17 will be intersecting. k−1 k−1 0 The condition S1 ∩ S2 ⊆ N [v] ∩ N [w] allows us again to conclude that s(T ) < s(T ).

3.4 Minimal vertex separators and strongly chordal graphs

We present this section as an application of the previous one. An analysis of the conditions stated in Theorem 3.17 for a graph to be dually chordal yields that none of them can be omitted in order to simplify the characterization. Some of the simplest graphs illustrating this are mentioned below. C4, the cycle of four vertices, satisﬁes that S(C4) is Helly and L(S(C4)) is chordal, but the minimal vertex separators of C4 or any longer cycle do not induce connected subgraphs.

33 Figure 3.5: The ﬁrst graphs of a family {Hn}n≥1 for which the number of minimal vertex separators grows exponentially.

We also know that the suns are not dually chordal graphs. Each one of their minimal vertex separators induces a connected subgraph. L(S(3 − sun)) is chordal, but S(3 − sun) is not Helly. On the other hand, for all k ≥ 4, S(k − sun) is Helly, but L(S(k − sun)) is not chordal. These examples are exactly the minimal forbidden induced subgraphs of the strongly chordal graphs. The current section is devoted to giving a new proof of this fact by using minimal vertex separators and the characterizations of dually chordal graphs from the previous section. The proof will be divided in two propositions, which are inspired in Propositions 1.18 and 1.19, and the main theorem at the end.

Lemma 3.19. Let S be a minimal vertex separator of a chordal graph G and v 6∈ S. Then, there exists a vertex w such that S ⊆ N[w] and v and w are in diﬀerent connected components of G − S.

Proof. Since S is a minimal vertex separator of a chordal graph, there exist cliques C1 and C2 in G such that C1 ∩ C2 = S and C1 \ C2 and C2 \ C1 are contained in diﬀerent connected components of G − S [16]. We can assume without loss of generality that C1 \ C2 and v are not contained in the same connected component of G − S. Then, any vertex w ∈ C1 \ C2 satisﬁes the required condition.

Lemma 3.20. [2] A family F is Helly if and only if, for every triple u, v, w of elements of F, the members of the subfamily Fuvw of members of F that contain at least two of the elements of the triple have a non-empty intersection.

Proposition 3.21. Let G be a chordal graph such that S(G) is not Helly. Then, the 3-sun is an induced subgraph of G.

Proof. Since S(G) is not Helly, we can take v1, v2, v3 ∈ V (G) such that the members of S(G) containing at least two elements of {v1, v2, v3} have empty intersection. Then, there exist S1,S2,S3 ∈ S(G) such that S1 ∩ {v1, v2, v3} = {v2, v3}, S2 ∩ {v1, v2, v3} = {v1, v3} and S3 ∩ {v1, v2, v3} = {v1, v2}. By Lemma 3.19, for i = 1, 2, 3, we can take a vertex wi such that Si ⊆ N[wi] and vi and wi are in diﬀerent connected components of G − Si. As the minimal vertex separators of G are complete, it is clear that {v1, v2, v3} is complete. Also, by our choice, N[w1] ∩ {v1, v2, v3} = {v2, v3}, N[w2] ∩ {v1, v2, v3} = {v1, v3} and N[w3] ∩ {v1, v2, v3} = {v1, v2}.

34 Now suppose that 1 ≤ i, j ≤ 3 and i 6= j. Then, wj 6∈ Si because vj ∈ Si and wj is not adjacent to it. Since wj is adjacent to vi, we can conclude that wi and wj are in diﬀerent connected components of G − Si. Thus, wi and wj are not adjacent. Therefore, {v1, v2, v3, w1, w2, w3} induces a 3-sun in G.

Proposition 3.22. Let G be a chordal graph such that S(G) is Helly and L(S(G)) is not chordal. Then, there exists a number k, k ≥ 4, such that G has an induced k-sun.

Proof. Let R be a chordless cycle of minimum length in L(S(G)) and S1S2...SkS1, k ≥ 4, be the sequence of vertices of R. Take vi ∈ Si ∩ Si+1, i = 1, ..., k − 1, and vk ∈ S1 ∩ Sk. Since R is chordless, all these vertices are different and form a cycle C in G. If the vertices of C are pairwise adjacent, we can pick, by Lemma 3.19, a vertex w1 such that S1 ⊆ N[w1] and such that w1 and {v1, v2, ..., vk}\{v1, vk} are contained in different connected components of G − S1. Similarly, for i = 2, ..., k, let wi be a vertex such that Si ⊆ N[wi] and wi and {v1, v2, ..., vk}\{vi−1, vi} are contained in different connected components of G − Si. Let 1 ≤ i, j ≤ k, i 6= j. It is not difficult to see that wj 6∈ Si and that wi and wj are in different connected components of G − Si, so wi is not adjacent to wj. Therefore, {v1, ..., vk, w1, ..., wk} induces a k-sun. If the vertices of C are not pairwise adjacent, take vi, vj in C such that vi and vj are not adjacent, and let S be a minimal vivj-separator. As there are two vivj-paths in C, S must contain two nonconsecutive vertices of C, let them be vl and vm. Without loss of generality, we may assume that l < m. Let A = {Sn : l + 1 ≤ n ≤ m}. If |A| ≤ 2, then it is clear that 0 0 S intersects every set in A. If |A| ≥ 3, let R be the cycle SSl+1 Sl+2...Sm S. Since R is shorter than R, R0 has a chord. As R is chordless, S must be one of the endpoints of the chord. The addition of this chord to R0 generates two new cycles. If any of these cycles has length at least four, then there is a chord, and again S must be one of the endpoints. We continue this procedure until concluding that the intersection between S and every element of A is not empty. Similarly, if we let B = {S1,S2, ..., Sk}\ A, we can also infer that S intersects every set in B. Since S(G) is Helly, we can take ui ∈ Si ∩ Si+1 ∩ S, i = 1, ..., k − 1, and uk ∈ S1 ∩ Sk ∩ S. Then, {u1, u2, ..., uk} is complete because it is a subset of S. We can find an induced k-sun as we did in the case that the vertices of C were pairwise adjacent.

Theorem 3.23. Let G be a graph. Then, G is a hereditary dually chordal graph if and only if G is chordal and without induced suns.

Proof. Since the cycles of length at least four and the suns are not dually chordal, none of them is an induced subgraph of a hereditary dually chordal graph. Now suppose that G is chordal and without induced suns. Then, every minimal vertex separator of G is a complete set, clearly inducing a connected subgraph. By Propositions 3.21 and 3.22, S(G) is Helly and L(S(G)) is chordal. Consequently, by Theorem 3.17, G is dually chordal. Every induced subgraph of G is also chordal and without induced suns, and hence dually chordal. Therefore, G is a hereditary dually chordal graph.

35 Chapter 4

Basic chordal graphs

Let G be a connected chordal graph. Then, we know that G has at least one clique tree. The set of vertices of each clique tree is C(G). Moreover, K(G) is a dually chordal graph and the set of vertices of each compatible tree of K(G) is V (K(G)), which equals C(G). Given that the clique trees of G and the compatible trees of K(G) have the same set of vertices, it is possible to compare them. In fact, we can see that they are closely related. As in the previous chapter, we assume that the graphs considered are always connected.

Proposition 4.1. Let G be a chordal graph. Then, every clique tree of G is compatible with K(G).

Proof. Let T be a clique tree of G and C be any clique of G. The closed neighborhood of C S in K(G) is equal to Cv, which induces a subtree because T is a clique tree. Therefore, T is v∈C compatible with K(G).

However, the converse is not necessarily true, that is, we can ﬁnd a chordal graph for which not all the compatible trees of its clique graph are clique trees. For example, consider the graph G of Figure 1.5. The compatible tree of K(G) shown there is not a clique tree of G because Cd does not induce a subtree (see Figure 4.1). A graph is deﬁned to be basic chordal if it is chordal and its clique trees are exactly the compatible trees of its clique graph. One of the main goals set for this chapter is to develop tools to answer as easily as possible whether a given chordal graph G is basic chordal or not. In view of Proposition 4.1, the problem reduces to determining if every tree compatible with K(G) is a clique tree of G. The main characterization of basic chordal graphs that is achieved appears in Theorem 4.26. In order to proceed, we need to know additional properties of chordal graphs, clique trees and compatible trees. First, we note that a clique tree of a given chordal graph can be found in polynomial time by using numerical algorithms. The most classical one relies on the following well known characterization:

Theorem 4.2. [18] Let G be a graph and K(G)w be the graph obtained from K(G) by assigning each edge CC0 the weight |C ∩ C0|. Then, T is a clique tree of G if and only if it is a maximum weight spanning tree of K(G)w of weight P |C| − |V (G)|. C∈C(G) Fortunately, there is a similar procedure to ﬁnd a compatible tree of a dually chordal graph.

36 Figure 4.1: A chordal graph G, its clique graph and a tree compatible with K(G) which is not a clique tree of G because Cd does not induce a subtree.

Theorem 4.3. [5] Let G be a graph and Gw be the same graph after assigning each edge uv the weight p(u, v) = |N[u] ∩ N[v]|. Then, a tree T is compatible with G if and only if it is a maximum weight spanning tree of Gw of weight 2|E(G)|.

Proof. Let T be a spanning tree of Gw. Then: X X X X p(u, v) = |N[u] ∩ N[v]| = |{w} ∩ N[u] ∩ N[v]| = uv∈E(T ) uv∈E(T ) uv∈E(T ) w∈V (G) X X X X |{w} ∩ N[u] ∩ N[v]| = |E(T [N[w]])| ≤ (|N[w]| − 1) = w∈V (G) uv∈E(T ) w∈V (G) w∈V (G) X X (1 + deg(w) − 1) = deg(w) = 2|E(G)| w∈V (G) w∈V (G) The equality holds if and only if, for all w ∈ V (G), |E(T [N[w]])| = |N[w]| − 1, that is, if and only if the closed neighborhood of every vertex of G induces a subtree of T .

The three properties that will follow are about minimal vertex separators, which are going to be useful to gain insight into the nature of the edges of clique trees. While the minimal vertex separators of the dually chordal graphs were the subject of the previous chapter, the minimal vertex separators of the chordal graphs will be among the main actors of this chapter. As the minimal vertex separators of chordal graphs are complete, it is a priori possible that some of them are cliques. The property that appears below shows that it is not the case. Given a graph G, two cliques C1 and C2 are a separating pair if C1 ∩ C2 separates any pair of vertices such that one is in C1 \ C2 and the other is in C2 \ C1. This deﬁnition implies that C1 ∩ C2 is a minimal vertex separator. It is also true that every minimal vertex separator of a chordal graph can be expressed in that way:

Theorem 4.4. [16] Let G be a chordal graph and S ∈ S(G). Then, there exists a separating pair C1, C2 such that S = C1 ∩ C2.

37 The importance of separating pairs lies in how they are related to clique trees. If we wonder what edges can be found in at least one clique tree, the following theorem gives the answer:

Theorem 4.5. [16] Let C1 and C2 be two distinct cliques of a chordal graph G. Then, there exists a clique tree T of G such that C1C2 ∈ E(T ) if and only if C1 and C2 form a separating pair.

Finally, it is interesting to note that, if just one clique tree of a graph is known, it is possible to determine what the edges of the other clique trees (if there is more than one) can be:

Theorem 4.6. [16] Let G be a chordal graph, T be a clique tree of G and C1,C2 ∈ C(G), C1 6= C2. Then, there exists a clique tree of G having C1C2 as an edge if and only if there are two cliques of G that are adjacent in T hC1,C2i and whose intersection equals C1 ∩ C2. Now we are about to establish a necessary and suﬃcient condition for a chordal graph not to be basic chordal. As an initial step, we deal with one previous result, which can be proved by using arguments similar to those that can demonstrate the eﬀectivity of some algorithms such as Kruskal’s [17].

Proposition 4.7. Let G be an edge-weighted graph and T , T 0 be two maximum weight spanning 0 trees of G. Then, there exists a sequence T1T2...Tk such that T1 = T , Tk = T , Ti is a maximum weight spanning tree of G for all 1 ≤ i ≤ k and, for all 2 ≤ i ≤ k, Ti can be obtained from Ti−1 by adding one edge of G to it and removing another.

Theorem 4.8. Let G be a chordal graph. Then, there exists a tree compatible with K(G) that is not a clique tree of G if and only if there exist S ∈ S(G) and C1,C2 ∈ C(G) such that C1 ∩C2 ( S and, for every C ∈ C(G), C ∩ S 6= ∅ implies that C ∩ C1 6= ∅ and C ∩ C2 6= ∅. Proof. Suppose that there exists a tree T that is compatible with K(G) but that is not a clique tree of G. Take a clique tree T 0 of G. Then, T 0 is compatible with K(G) because of Proposition 4.1 and, as a consequence of Theorem 4.3 and Proposition 4.7, there is a sequence 0 of trees compatible with K(G), T1T2...Tk, T1 = T , Tk = T , such that every tree of the sequence diﬀerent from T1 is built from its predecessor by adding one edge and removing another. Let i, 1 ≤ i ≤ k−1, be a number such that Ti is a clique tree of G and Ti+1 is not. Let C1C2 be the edge that is added and C3C4 be the edge that is removed to get Ti+1 from Ti. Then, C3,C4 ∈ Ti[C1,C2] (see Figure 4.2), which implies that NK(G)[C1] ∩ NK(G)[C2] ⊆ NK(G)[C3] ∩ NK(G)[C4]. We also infer from Theorem 4.3 that |NK(G)[C1] ∩ NK(G)[C2]| = |NK(G)[C3] ∩ NK(G)[C4]| and thus the two intersections are equal (∗). Since Ti is a clique tree, C1 ∩ C2 ⊆ C3 ∩ C4. But, as Ti+1 is not a clique tree, we infer from Theorem 4.2 that |C1 ∩ C2| < |C3 ∩ C4|. Set S = C3 ∩ C4. Then, by Theorem 4.5, S ∈ S(G), C1 ∩ C2 ( S and the condition that, for every C ∈ C(G), C ∩ S 6= ∅ implies C ∩ C1 6= ∅ and C ∩ C2 6= ∅ is deduced from (∗). Conversely, suppose that there exist S ∈ S(G) and C1,C2 ∈ C(G) such that C1 ∩C2 ( S and, for every C ∈ C(G), C ∩ S 6= ∅ implies C ∩ C1 6= ∅ and C ∩ C2 6= ∅. Let C3,C4 be a separating pair of G such that C3 ∩ C4 = S and T be a clique tree of G such that C3C4 ∈ E(T ). Consider the following cases: (1) C3,C4 ∈ T [C1,C2]: the hypothesis and the fact that T is compatible with K(G) imply that NK(G)[C1] ∩ NK(G)[C2] = NK(G)[C3] ∩ NK(G)[C4]. Then, T + C1C2 − C3C4 is compatible with K(G) but, as |C1 ∩ C2| < |C3 ∩ C4|, it is not a clique tree of G. (2) Otherwise, C1 and C2 are in the same connected component of T − C3C4. This implies that C3 ∈ T [C1,C4] ∩ T [C2,C4] or that C4 ∈ T [C1,C3] ∩ T [C2,C3] (see Figure 4.3).

38 Figure 4.2: If Ti + C1C2 − C3C4 is a tree, then C3,C4 ∈ Ti[C1,C2].

Figure 4.3: If C1 and C2 are in the same connected component of T − C3C4 as C3, then C3 ∈ T [C1,C4] ∩ T [C2,C4] (left). If they are in the same connected component as C4, then C4 ∈ T [C1,C3] ∩ T [C2,C3] (right).

Suppose that C3 ∈ T [C1,C4] ∩ T [C2,C4]. Then, C1 ∩ C4 ⊆ C3 ∩ C4 and C2 ∩ C4 ⊆ C3 ∩ C4. 0 If C1 ∩ C4 = C3 ∩ C4 and C2 ∩ C4 = C3 ∩ C4, then T := T + C1C4 − C3C4 is a clique tree of 0 G. Since C1 ∈ T [C2,C4], C2 ∩ C4 ⊆ C1 ∩ C2. This implies that C3 ∩ C4 ⊆ C1 ∩ C2, which is a contradiction. Consequently, C1 ∩ C4 ( C3 ∩ C4 or C2 ∩ C4 ( C3 ∩ C4. Furthermore, for every C ∈ C(G), C ∩ S 6= ∅ implies that C ∩ C1 6= ∅, C ∩ C2 6= ∅, C ∩ C3 6= ∅ and C ∩ C4 6= ∅. If C1 ∩ C4 ( C3 ∩ C4, then the fact that C3,C4 ∈ T [C1,C4] implies that case (1) can be applied with C1,C4 instead of C1,C2. The case that C2 ∩ C4 ( C3 ∩ C4 is analogous. If C4 ∈ T [C1,C3] ∩ T [C2,C3], then the proof is similar.

4.1 Subtree inducing sets and the concept of basis

Considering that clique trees and compatible trees are characterized by the fact that some particular sets induce subtrees of them, it is natural to wonder what other sets induce subtrees as well. This question is now going to be studied and the conclusions obtained will prove to be useful to continue learning about basic chordal graphs. Let G be a graph. If G is chordal, then SC(G) will denote the family of all sets F such that, for every clique tree T of G, T [F ] is a subtree of T . For instance, every member of the

39 dual clique family of G is in SC(G), which is a consequence of the definition of clique tree. By Proposition 3.2, SC(G) is a Helly family. Similarly, if G is dually chordal, SDC(G) will denote the family of all sets F such that, for every T compatible with G, T [F ] is a subtree of T . This family is Helly as well and, among the most known members of SDC(G), are the cliques, closed neighborhoods and minimal vertex separators of G. The latter is a consequence of the characterizations of compatible trees that we already know. These definitions and Proposition 4.1 imply that, for a chordal graph G, SDC(K(G)) ⊆ SC(G). It is not always easy to list all the members of SC(G) or SDC(G) because in general there is no polynomial bound for the cardinality of these families. But it would be desirable to know a procedure that would generate them all in case that only 88a few00 members are known. Given a family F of sets, the union S F is said to be connected if the intersection graph F ∈F c of F is connected. This will be denoted by S F . It is not difficult to see that families such as F ∈F SC(G) and SDC(G) are closed under intersections and connected unions. Suppose that a family F is closed under connected unions. Call a subfamily B of F generating if every member of F with more than one element can be expressed as the connected union of some members of B. The subfamily B is a basis for F if it is generating and no proper subfamily of B generates F. One condition to determine whether one member of a family is in some basis is the following:

Proposition 4.9. Let B be a basis for F and D ∈ F, |D| > 1. The following are equivalent:

(a) D is in B.

c (b) For every subfamily F 0 of F, D = S F implies that there exists F ∈ F 0 such that F = D. F ∈F0

Proof. c (a) ⇒ (b): Let F 0 be a subfamily of F such that D = S F be the non-unit members of F ∈F0 c 0 S F , so D = Fi as well. For each value of i between 1 and n, let Di1, ..., Dim be members of 1≤i≤n c c S S B such that Fi = Dij. Consequently, D = Dij. If no Dij is equal to D, then the 1≤j≤m 1 ≤ i ≤ n 1 ≤ j ≤ m last equality would imply that B\{D} also generates F, contradicting that B is a basis. So let i, j, 1 ≤ i ≤ n, 1 ≤ j ≤ m, be such that Dij = D. Since Dij ⊆ Fi ⊆ D, Fi = D. c S (b) ⇒ (a): As B is a basis, let D1, ..., Dn be members of B such that D = Di. By our 1≤i≤n assumption, Di = D for some value of i. Hence, D ∈ B. As condition (b) is independent of the basis that we are considering, the following conclusion is immediate:

Corollary 4.10. F has a unique basis.

40 In the following, the major results of this section are stated and proved. First, we show the relationship between the two families that have just been deﬁned and basic chordal graphs. Then, given a chordal graph G, the basis of SC(G) is found.

Theorem 4.11. Let G be a chordal graph. The following statements are equivalent:

(a) Every tree compatible with K(G) is a clique tree of G.

(b) SC(G) = SDC(K(G)).

Proof. (a) ⇒ (b): Proposition 4.1 and (a) imply that, for every tree T , T is a clique tree of G if and only if it is compatible with K(G). Therefore, the deﬁnitions of SC(G) and SDC(K(G)) imply that these families are equal. (b) ⇒ (a): The equality between SC(G) and SDC(K(G)) implies that DC(G) ⊆ SDC(K(G)). Therefore, every tree compatible with K(G) is a clique tree of G. (b) ⇔ (c): Trivial.

The name basic chordal for the class of graphs that is being studied was inspired by part (c) of Theorem 4.11.

Proposition 4.12. Let G be a chordal graph and A ∈ SC(G). If C1,C2 is a separating pair contained in A, then CC1∩C2 ⊆ A.

Proof. Let C be any element of CC1∩C2 . If C = C1 or C = C2, then it is clear that C ∈ A. Now suppose that C 6= C1 and C 6= C2. Let T be a clique tree of G such that C1C2 ∈ E(T ). We can also assume without loss of generality that C2 ∈ T [C,C1], so C ∩ C1 ⊆ C1 ∩ C2. As C ∈ CC1∩C2 , 0 C1 ∩ C2 ⊆ C ∩ C1, and we infer from Theorem 4.2 that T := T − C1C2 + CC1 is a clique tree 0 0 0 of G. Since A ∈ SC(G), T [A] is a subtree of T . Furthermore, C ∈ T [C1,C2] and C1,C2 ∈ A. Thus, C ∈ A.

The inclusion CC1∩C2 ⊆ A follows.

Theorem 4.13. Let G be a chordal graph. Then, {CS,S ∈ S(G)} is the basis of SC(G). Proof. Let A be any member of SC(G) with |A| > 1 and T be a clique tree of G. Denote by e1, e2, ..., ek all the edges of T [A] and by Si, i = 1, ..., k, the minimal vertex separator of G equal to the intersection of the endpoints of ei. By Proposition 4.12, CSi ⊆ A, i = 1, ..., k. This implies k S that CSi ⊆ A. As it is also true that CSi contains the endpoints of ei, i = 1, ..., k, and that i=1 k c S S T [A] is a subtree, we infer that A ⊆ CSi . Therefore, A = CSi . i=1 1≤i≤k Hence, {CS,S ∈ S(G)} is a generating subfamily of SC(G). Now, let S be a ﬁxed minimal vertex separator and C1C2 be any edge of T such that c S C1 ∩ C2 = S. Suppose that {M1, .., Mn} is a subfamily of SC(G) such that CS = Mi and let 1≤i≤n T1 and T2 be the subtrees we obtain by removing the edge C1C2 from T [CS], with C1 ∈ V (T1) n S and C2 ∈ V (T2). As Mi is connected, there must be an index j, 1 ≤ j ≤ n, such that i=1

41 Mj ∩ V (T1) 6= ∅ and Mj ∩ V (T2) 6= ∅. Since T [Mj] is a subtree of T [CS], C1,C2 ∈ Mj. Now we apply Proposition 4.12 to get that CC1∩C2 ⊆ Mj, that is, CS ⊆ Mj. Hence, CS = Mj. Thus, by Proposition 4.9, CS is a member of the basis of SC(G). Therefore, {CS,S ∈ S(G)} is the basis of SC(G).

By noting that the number of members of C(G) and S(G) is of order O(|V (G)|) because G is chordal, we conclude that computing the basis of SC(G) can be done eﬃciently in polynomial time. If we call dimension of G to the number of members of the basis of SC(G), then Theorem 4.13 implies that the dimension equals |S(G)|, which, by Theorems 4.4, 4.5 and 4.6, is at most the number of cliques minus one. On the other hand, if G is dually chordal, deﬁne the dual dimension of G as the number of members of the basis of SDC(G). It is clear from Theorem 4.11 that, if G is basic chordal, the dimension of G is equal to the dual dimension of K(G). But the converse is not necessarily true.

Figure 4.4: A chordal graph G, its clique graph and a tree T that is compatible with K(G) but is not a clique tree of G.

Consider the chordal graph G of Figure 4.4. Its cliques are C1 = {2, 4, 5}, C2 = {1, 2, 3}, C3 = {2, 3, 5, 6} and C4 = {5, 6, 7}. The separating pairs are C1,C3; C2,C3 and C3,C4, so S(G) = {{2, 3}, {2, 5}, {5, 6}}. By Theorem 4.13, the basis of SC(G) is {{C1,C3}, {C2,C3}, {C3,C4}}. Due to the simplicity of K(G), it is not diﬃcult to verify that the basis of SDC(K(G)) is {{C1,C2,C3}, {C1,C3}, {C1,C3,C4}}. In this case, both the dimension of G and the dual dimension of K(G) equal 3, but the bases are diﬀerent, so there is at least one tree compatible with K(G) that is not a clique tree of G. One example is T , also displayed in Figure 4.4. It is not a clique tree because neither C3 nor C6 induce subtrees of T .

4.2 More results on basic chordal graphs and dually chordal graphs

In this section, we continue developing the necessary work to ﬁnd a new characterization of basic chordal graphs. Given a dually chordal graph G, we ﬁrst characterize in Theorem 4.15 all the basic chordal graphs having clique graph equal to G. This will allow to prove several properties of dually chordal graphs. Particularly, we will obtain a new characterization of dually

42 chordal graphs based on compatible trees (Theorem 4.22), and the basis of SDC(G) will be found (Theorem 4.18). Finally, knowing how to ﬁnd the basis when we work both on chordal and dually chordal graphs will give us a characterization of basic chordal graphs which is then applied to prove a couple of additional properties of basic chordal graphs.

We will say that a chordal graph H is in correspondence with G if H is basic chordal and K(H) = G. The next steps are to ﬁnd all chordal graphs in correspondence with G.

Proposition 4.14. [14] Let F be a Helly separating family. Then, C(L(F)) = DF.

Theorem 4.15. Let G be a dually chordal graph and H be a chordal graph. Then, H is in correspondence with G if and only if H is the intersection graph of a separating subfamily of SDC(G) whose two section is equal to G.

Proof. Suppose that H is in correspondence with G. We know that H is equal to the intersection graph of {Cv}v∈V (H), which is a subfamily of SC(H). It is not diﬃcult to verify that the two section of this family is K(H), which equals G, and that it is separating. Furthermore, by Theorem 4.11, SC(H) = SDC(K(H)) = SDC(G), so {Cv}v∈V (H) ⊆ SDC(G). Conversely, assume that H is the intersection graph of a separating family F such that S(F) = G and all its members belong to SDC(G). By Theorem 1.4, H is chordal. Now we show that K(H) is isomorphic to G. Since F is Helly and separating, we apply Proposition 4.14 to obtain that C(H) = C(L(F)) = DF. In addition, for all u, v ∈ V (G),

DuDv ∈ E(K(H)) ⇔ Du ∩ Dv 6= ∅ ⇔ ∃F ∈ F,F ∈ Du ∧ F ∈ Dv ⇔ ∃F ∈ F, u ∈ F ∧ v ∈ F ⇔ uv ∈ E(G) where the last equivalence is justiﬁed by the fact that the two section of F is G. Thus, the function f : V (G) → V (K(H)) such that f(v) = Dv is a graph isomorphism between G and K(H). For every F ∈ F, consider the member CF of DC(H). Then, CF = {C ∈ C(H): F ∈ C} = {Dv ∈ DF : v ∈ F }. As F ∈ SDC(G), {Dv ∈ DF : v ∈ F } ∈ SDC(K(H)). Consequently, DC(H) ⊆ SDC(K(H)). Therefore, every tree compatible with K(H) is a clique tree of H, which completes the proof.

As an example of this theorem, consider the graph K(G) of Figure 4.4. We know that G is not in correspondence with K(G). Let us look for a graph in correspondence with K(G). {C1,C2,C3} and {C1,C3,C4} form a subfamily of SDC(K(G)) whose two section equals K(G), but it is not separating. Then, let F = {{C1}, {C2}, {C3}, {C4}, {C1,C2,C3}, {C1,C3,C4}}. By Theorem 4.15, the intersection graph of this family (see Figure 4.5) is in correspondence with K(G). Theorem 4.15 provides the ideal framework so that most of the properties that have been seen in the beginning of the chapter can be used to obtain properties of compatible trees similar to those of clique trees and properties about some other things.

Theorem 4.16. Let G be a dually chordal graph, F be a separating subfamily of SDC(G) such that S(F) = G and DF = {Dv}v∈V (G). Then: (a) Given u, v ∈ V (G), there exists a tree T compatible with G and such that uv ∈ E(T ) if and only if Du and Dv form a separating pair of L(F).

43 Figure 4.5: The dually chordal graph K(G) of Figure 4.4 and, to the right, a chordal graph whose clique trees are exactly the compatible trees of K(G).

(b) If T is a tree compatible with G, there exists a tree T 0 compatible with G such that uv ∈ E(T 0) if and only if there are two vertices x and y adjacent in T hu, vi such that Dx∩Dy = Du∩Dv. w (c) Assign to each uv ∈ E(G) the number |Du ∩ Dv| to obtain the weighted graph G . Then, a tree T is compatible with G if and only if it is a maximum weight spanning tree of Gw of weight P |F | − |F|. F ∈F

(d) If T is compatible with G, A ∈ SDC(G), uv ∈ E(T ) and {u, v} ⊆ A, then T F ⊆ A. F ∈Du∩Dv T (e) If T is compatible with G, then F :: uv ∈ E(T ) is the basis of SDC(G). F ∈D ∩D F ∈Du ∩Dvv Proof. By Proposition 4.14, C(L(F)) = DF. Then: (a) Apply Theorem 4.15 and then use Theorem 4.5 on L(F). (b) Apply Theorem 4.15 and then use Theorem 4.6 on L(F). (c) Apply Theorem 4.15 and then use Theorem 4.2 on L(F), noting that X X |C| − |V (L(F))| = |Dv| − |F| = C∈C(L(F)) v∈V (G) X X X X X |{v} ∩ F | − |F| = |{v} ∩ F | − |F| = |F | − |F| v∈V (G) F ∈F F ∈F v∈V (G) F ∈F (d) Apply part (a), and Theorems 4.15 and 4.11 and Proposition 4.12 on L(F), noting that

CDu∩Dv = {C ∈ C(L(F)) : Du ∩ Dv ⊆ C} = {Dw : Du ∩ Dv ⊆ Dw} = T Dw : w ∈ T F {Dw : ∀F ∈ F,F ∈ Du ∩ Dv → w ∈ F } = Dw : w ∈ F FF∈∈DDuu∩∩DDvv TT (e) Part (d) implies that FF :: uvuv ∈∈ EE((TT)) generates SDC(G). The minimality FF∈∈DDuu∩∩DDvv is a consequence of part (a) and Theorems 4.15, 4.11 and 4.13 applied to L(F).

44 The most typical example of a family with the characteristics mentioned in Theorem 4.16 is the one consisting of the cliques of G and the unit sets of vertices. Applying some of the results of Theorem 4.16 leads to the following conclusions: Theorem 4.17. Let G be a dually chordal graph. Assign to each edge uv ∈ E(G) the number w |Cu ∩ Cv| to obtain the weighted graph G . Then, a tree T is compatible with G if and only if it is a maximum weight spanning tree of Gw of weight P |C| − |C(G)|. C∈C(G) Proof. Set F = C(G) ∪ {{v} : v ∈ V (G)}. Apply part (c) of Theorem 4.16 to this family and note that, for every uv ∈ E(G), |Du ∩ Dv| = |Cu ∩ Cv| and

X X X |F | − |F| = |C| + |V (G)| − (|C(G)| + |V (G)|) = |C| − |C(G)| F ∈F C∈C(G) C∈C(G)

Theorem 4.17 has long been known [13] and has been proved independently by many authors. Theorem 4.18. Let G be a dually chordal graph and T be compatible with G. Then, T C : uv ∈ E(T ) is the basis of SDC(G). C∈C ∩C C∈Cu∩Cv Proof. Apply part (e) of Theorem 4.16 to F deﬁned as in the previous theorem. Note again that Du ∩ Dv = Cu ∩ Cv.

Figure 4.6: A dually chordal graph and a tree compatible with it.

As an example, consider again the graph G of Figure 4.4. It is also a dually chordal graph and one of its compatible trees appears in Figure 4.6. Now we use the edges of that tree to obtain the basis of SDC(G). Consider the edge 12 of T . The only clique containing 1 and 2 is {1, 2, 3}, so this set is in the basis. Now consider the edge 23. The cliques that contain both 2 and 3 are {1, 2, 3} and {2, 3, 5, 6}. Their intersection equals {2, 3}. Thus, this set is also in he basis. Similarly, if we consider the remaining edges of the compatible tree, we get that the other members of the basis are {2, 5}, {5, 6}, {2, 4, 5} and {5, 6, 7}. Computing the basis was easy for the graph in the example because it is small, but it could become diﬃcult to compute the basis for larger graphs as there is no polynomial bound

45 for the number of cliques of an arbitrary dually chordal graph. For example, given a positive integer n, deﬁne nK2 as the graph with n connected components, each having two vertices. The n complement of nK2 is a graph with 2 cliques, and the graph obtained from nK2 by adding a universal vertex is a dually chordal graph with the same number of cliques. Nonetheless, a connection with neighborhoods reveals that it is simpler than we would expect:

Proposition 4.19. Let u and v be adjacent vertices of a graph G. Then, T C = C∈Cu∩Cv T N[w]. w∈N[u]∩N[v] Proof. We prove that the two sets include each other. Let x ∈ T C and w ∈ N[u] ∩ N[v]. Then, {u, v, w} is a complete set and there exists a C∈Cu∩Cv clique C1 such that {u, v, w} ⊆ C1. This implies that C1 ∈ Cu ∩Cv, and hence x ∈ C1. Therefore, x ∈ N[w]. We can conclude that T C ⊆ T N[w]. C∈C ∩C w∈N[u]∩N[v] T u v Conversely, let x ∈ N[w] and C ∈ Cu ∩ Cv. Then, C ⊆ N[u] ∩ N[v] and, by the w∈N[u]∩N[v] description of x, x is in the closed neighborhood of every element of C. As a consequence, x ∈ C. We can infer from this reasoning that T N[w] ⊆ T C. w∈N[u]∩N[v] C∈Cu∩Cv

Another alternative for computing the basis of SDC(G) is to replace C(G) by a subfamily F whose members are selected as follows: for each edge uv of G, pick C ∈ C(G) such that {u, v} ⊆ C. It is straightforward that |F| ≤ |E(G)| and that S(F) = C(G).

Now we are going to use the last two results to obtain a characterization of dually chordal graphs generalizing the one saying that a graph is dually chordal if and only if there exists a spanning tree T such that every clique of the graph induces a subtree T . We say that a set A of vertices of a graph G is positive boolean if it can be obtained by repeated intersections and unions of closed neighborhoods. It is connected positive boolean if connected unions are used instead of common unions. A combination of Theorem 4.18 and Proposition 4.19 reveals that the members of the basis of SDC(G), being G dually chordal, are connected positive boolean, and so will the connected unions of them be. Since the closed neighborhood of every vertex induces a subtree of every compatible tree of the graph, we can conclude:

Theorem 4.20. Let G be a dually chordal graph and A ⊆ V (G). Then, A ∈ SDC(G) if and only if it is connected positive boolean.

The following two results show what the new characterization of dually chordal graphs is.

Theorem 4.21. Let G be a graph, T be a spanning tree of G and F be a family of connected positive boolean subsets of V (G) such that S(F) = G. Then, the following statements are equivalent:

(a) T is compatible with G.

(b) Every member of F induces a subtree of T .

46 Proof. (a) ⇒ (b): Since T is compatible with G, this graph is dually chordal. By Theorem 4.20, every member of F is in SDC(G), so it induces a subtree of T . (b) ⇒ (a): Let u and v be two adjacent vertices of G and w ∈ T (u, v). Since S(F) = G, we can take F ∈ F such that {u, v} ⊆ F . As T [F ] is a subtree of T , w ∈ F . Therefore, w is adjacent to u and v in S(F), that is, uw ∈ E(G) and vw ∈ E(G). Consequently, by Theorem 1.15, T is compatible with G.

Corollary 4.22. Let G be a graph and F be a family of connected positive boolean subsets of V (G) such that S(F) = G. Then, G is dually chordal if and only if there exists a spanning tree T of G such that every member of F induces a subtree of T .

We now resume the study of basic chordal graphs. As we know how the basis of SDC(G) can be calculated, we can get a new characterization of basic chordal graphs. Its statement is preceded by some lemmas.

Lemma 4.23. Let G be a chordal graph and C1,C2 be a separating pair of G. If C is a clique such that C ∩ C1 6= ∅ and C ∩ C2 6= ∅, then C ∩ C1 ∩ C2 6= ∅.

Proof. Let T be a clique tree of G such that C1C2 ∈ E(T ). Then, C1 ∈ T [C,C2] or C2 ∈ T [C,C1]. In the ﬁrst case, C ∩ C2 ⊆ C1 ∩ C2. In the second, C ∩ C1 ⊆ C1 ∩ C2. In either case, C ∩ C1 ∩ C2 6= ∅. Suppose now that G is a dually chordal graph and that we have a compatible tree T with a given order for its edges. If we look at (e) in Theorem 4.16, it could happen that, depending on the family F that is used, the members of the basis of SDC(G) can be obtained in diﬀerent possible orders. It is not the case, as the following lemma shows. Lemma 4.24. Let G be a dually chordal graph, T be a tree compatible with G, uv ∈ E(T ) and T F be any separating subfamily of SDC(G) such that S(F) = G. Then, BF := F does F ∈Du∩Dv not depend on the choice of F. Proof. Let F and F 0 be separating subfamilies of SDC(G) such that the two section of each one equals G. The set {u, v} is contained in both BF and BF 0 , so part (d) of Theorem 4.16 0 can be applied to both F and F to conclude that BF ⊆ BF 0 and BF 0 ⊆ BF , respectively. Consequently, BF = BF 0 .

Given a graph G and S ∈ S(G), deﬁne BS as the set of cliques of G that intersect every T clique intersecting S, i.e., BS = NK(G)[C]. This set has not to be mistaken for BF from C∩S6=∅ the previous lemma. Now we prove:

Lemma 4.25. Let G be a chordal graph. Then, {BS : S ∈ S(G)} is the basis of SDC(K(G)). Proof. Let S be any minimal vertex separator of G and C ,C be a separating pair such that T 1 2 T S = C1 ∩ C2. Then, by Lemma 4.23, NK(G)[C] = NK(G)[C]. Now C∩S6=∅ C∈NK(G)[C1]∩NK(G)[C2] T apply Proposition 4.19 to conclude that BS = D. D ∈ C(K(G)) C1,C2 ∈ D Finally, take a clique tree T for G and complete the proof by applying Proposition 4.1 and Theorems 4.4, 4.5, 4.6 and 4.18.

47 If G is a basic chordal graph, then we deduce from Theorems 4.11 and 4.13 and Lemma 4.25 0 that {BS : S ∈ S(G)} = {CS : S ∈ S(G)}. Thus, for each S ∈ S(G), there exists S ∈ S(G) 0 such that BS = CS0 . In principle, S does not need to equal S. However, the equality always holds, leading to the following characterization: Theorem 4.26. Let G be a chordal graph. Then, G is basic chordal if and only if, for every S ∈ S(G), BS = CS.

Proof. Suppose that, for every S ∈ S(G), BS = CS. Therefore, by Theorem 4.13 and Lemma 4.25, the bases for SC(G) and SDC(K(G)) are equal. It follows from Theorem 4.11 that G is basic chordal. Conversely, suppose that G is a basic chordal graph. Let S ∈ S(G), T be a clique tree of S G and C1C2 ∈ E(T ) be such that C1 ∩ C2 = S. Set F1 = C(K(G)) {{C} : C ∈ C(G)} and F2 = {Cv : v ∈ V (G)}. Use the expression found for BS in Lemma 4.25 and apply Lemma 4.24 to C1,C2, F1, F2 to get that \ \ \ \ BS = D = BF1 = BF2 = Cv = Cv = Cv = CS D ∈ C(K(G)) C1,C2∈Cv v∈C1∩C2 v∈S C1,C2 ∈ D

Note that, as we know that the number of minimal vertex separators does not exceed the number of vertices in chordal graphs, testing this condition proves viable in terms of algorithmic complexity. Many of the results and proofs about chordal graphs in this chapter tacitly assume that the graphs are not complete, otherwise the family of minimal vertex separators is empty. However, it is easy to see that the theorems and propositions keep being true for the case of complete graphs. Although Theorem 4.26 was discovered thanks to basis theory, going back to Theorem 4.8 reveals that this result could have also been used to give a simple proof of Theorem 4.26. Here it is shown how: Theorem 4.27. Let G be a chordal graph. The following are equivalent:

1. There exist S ∈ S(G) and C1,C2 ∈ C(G) such that C1 ∩ C2 ( S and, for all C ∈ C(G), C ∩ S 6= ∅ implies that C ∩ C1 6= ∅ and C ∩ C2 6= ∅.

2. There exists S ∈ S(G) such that BS 6= CS.

Proof. Suppose that 1. is true and let C1,C2,S have the characteristics mentioned there. We shall prove that BS 6= CS. Since C1 ∩C2 ( S, it is impossible that S ⊆ C1 and that S ⊆ C2 at the same time, otherwise S ⊆ C1 ∩ C2. Suppose without loss of generality that S is not contained in C1. Then, C1 6∈ CS but, by the hypothesis, C1 ∈ BS. Therefore, BS 6= CS. Conversely, suppose that 2. is true and take S ∈ S(G) such that BS 6= CS. Since CS ⊆ BS, there exists D ∈ C(G) such that D ∈ BS and D 6∈ CS. Let T be a clique tree of G and C3,C4 be such that C3C4 ∈ E(T ) and C3 ∩ C4 = S. Then, C4 ∈ T [C3,D] or C3 ∈ T [C4,D]. Suppose without loss of generality that C4 ∈ T [C3,D]. We infer from Theorem 1.5 that C3 ∩ D ⊆ C3 ∩ C4 = S. If C3 ∩ D = S, then S ⊆ D, which contradicts that D 6∈ CS. Therefore, C3 ∩ D ( S.

48 If C is any clique of G such that C ∩ S 6= ∅, then C ∩ C3 6= ∅ because S ⊆ C3, and C ∩ D 6= ∅ because D ∈ BS. Therefore, we can set C1 = C3 and C2 = D to verify that 1. is true. As an example of Theorem 4.26, consider the two graphs of Figure 4.7.

Figure 4.7: Every tree compatible with K(G0) is a clique tree of G0. The same cannot be said about G.

K(G) equals the complete graph on four vertices, implying that every spanning tree of K(G) is a compatible tree. However, we know that the suns have only one clique tree. Therefore, G is not basic chordal. In order to conclude the same through Theorem 4.26, note that S(G) = {{2, 3}, {2, 5}, {3, 5}} and C(G) = {{1, 2, 3}, {2, 3, 5}, {2, 4, 5}, {3, 5, 6}}. Then, the comparison between the sets BS and CS, for each S ∈ S(G), is given by the following table:

S BS CS BS = CS {2, 3} C(G) {{1, 2, 3}, {2, 3, 5}} × {2, 5} C(G) {{2, 3, 5}, {2, 4, 5}} × {3, 5} C(G) {{2, 3, 5}, {3, 5, 6}} ×

But if we just add three new vertices of degree one to G to get G0, the result is diﬀerent. This time, C(G0) = C(G) S {{2, 7}, {3, 8}, {5, 9}} and S(G0) = S(G) S {{2}, {3}, {5}}. The corresponding table is:

S BS CS BS = CS {2, 3} {{1, 2, 3}, {2, 3, 5}} {{1, 2, 3}, {2, 3, 5}} X {2, 5} {{2, 3, 5}, {2, 4, 5}} {{2, 3, 5}, {2, 4, 5}} X {3, 5} {{2, 3, 5}, {3, 5, 6}} {{2, 3, 5}, {3, 5, 6}} X {2} {{1, 2, 3}, {2, 7}, {2, 3, 5}, {2, 4, 5}} {{1, 2, 3}, {2, 7}, {2, 3, 5}, {2, 4, 5}} X {3} {{1, 2, 3}, {3, 8}, {2, 3, 5}, {3, 5, 6}} {{1, 2, 3}, {3, 8}, {2, 3, 5}, {3, 5, 6}} X {5} {{2, 4, 5}, {5, 9}, {2, 3, 5}, {3, 5, 6}} {{2, 4, 5}, {5, 9}, {2, 3, 5}, {3, 5, 6}} X This clearly shows that the class of basic chordal graphs is not hereditary. In fact, a construction like the one used to obtain the graph G0 allows to show that every chordal graph is the induced subgraph of some basic chordal graph.

49 Proposition 4.28. Let G be a chordal graph and V 0 be the set of vertices of G that are not simplicial. Let G0 be the graph constructed from G by adding, for each v ∈ V 0, a vertex v∗ and the edge vv∗. Then, G0 is basic chordal.

Proof. If V 0 = ∅, then G is a complete graph, G = G0 and it is easy to check that G is basic chordal. If V 0 6= ∅, then S(G0) = S(G) S {{v} : v ∈ V 0}. We now prove that, for every 0 0 0 S ∈ S(G ), BS = CS, where the two sets are computed with respect to G . Let S ∈ S(G ), C ∈ BS and v ∈ S. As v is the vertex of a minimal vertex separator, v is not simplicial and 0 ∗ 0 ∗ hence v ∈ V . Then, {v, v } is a clique of G intersecting S. Since C ∈ BS, C ∩ {v, v }= 6 ∅. Thus, v ∈ C. We can conclude from this reasoning that S ⊆ C, that is, C ∈ CS. Therefore, BS ⊆ CS. By the deﬁnitions, the inclusion CS ⊆ BS is always true. Then, BS = CS. By Theorem 4.26, G0 is basic chordal.

We ﬁnish the section by stating a property relating basic chordal graphs to metric concepts.

Proposition 4.29. Let G be a chordal graph such that diam(G) ≤ 2. Then, G is basic chordal if and only if every vertex of G is either simplicial or universal.

Proof. Suppose that every vertex of G is simplicial or universal. If G is complete, then it is straightforward that G is basic chordal. If G is not complete, then the intersection of every two distinct cliques of G is equal to the set of universal vertices of G. Thus, by Theorem 4.4, the set of universal vertices of G is the only minimal vertex separator of G. Let S denote this separator. It is easy to verify that BS = CS = C(G). Therefore, by Theorem 4.26, G is basic chordal. Conversely, suppose that G is basic chordal. Now we show that K(G) is complete. This is direct in case that G is complete. Otherwise, let C1 and C2 be any two distinct cliques of G. We will prove that C1 ∩ C2 6= ∅. Let T be a clique tree of G. If C1 and C2 are adjacent in T , then it is clear that they are not disjoint. If they are not adjacent in T , let C ∈ T (C1,C2). Let also C3 be a leaf of T such that C1 ∈ T [C,C3] and C4 be another leaf such that C2 ∈ T [C,C4]. Then, C3 and C4 are simplicial cliques of G. Let v and w be simplicial vertices in C3 and C4, respectively. Since diam(G) ≤ 2, N[v] ∩ N[w] 6= ∅, that is, C3 ∩ C4 6= ∅. By the construction, {C1,C2} ⊆ T [C3,C4]. Thus, by Proposition 1.5, C3 ∩ C4 ⊆ C1 ∩ C2. Therefore, C1 ∩ C2 6= ∅. As a consequence, K(G) is a complete graph. Hence, every spanning tree of K(G) is a compatible tree. As G is basic chordal, we infer that every spanning tree of K(G) is a clique tree of G. It is easily deduced from the last fact that the members of SC(G) are C(G) and its unit subsets. Let v be a vertex of G. Thus, Cv ∈ SC(G). If Cv = C(G), then v is a universal vertex of G. If Cv is a unit set, then v is simplicial. This concludes the proof.

4.3 Determining possible sets of leaves for the compatible trees of a dually chordal graph

This short section is presented as an application of the previous one. Once again, the idea will be to take advantage of knowing how to solve certain problems about the clique trees of the chordal graphs with the purpose of solving problems about the compatible trees of the dually chordal graphs. For a tree T , the set of leaves of T will be denoted by L(T ).

50 As the title indicates, the problem will be as follows: given a set A of vertices of a dually chordal graph G, determine whether G has a compatible tree T such that L(T ) = A. In order to orient ourselves in the process of ﬁnding a solution, a necessary condition is obtained ﬁrst. Our interest will be in graphs with at least three vertices, because the problem is trivial on graphs with fewer vertices. Proposition 4.30. Let G be a dually chordal graph and T be a tree compatible with G. If v is a leaf of T and w is the vertex such that vw ∈ E(T ), then N[v] ⊆ N[w]. Proof. It holds, for every vertex u in N[v] \{v, w}, that w ∈ T (u, v). We infer from Theorem 1.15 that w is adjacent to u. Thus, N[v] \{v, w} ⊆ N[w]. As {v, w} is also a subset of N[w], the inclusion N[v] ⊆ N[w] follows.

Corollary 4.31. Let G be a dually chordal graph, |V (G)| ≥ 3, and T be a tree compatible with G. Then, every vertex in L(T ) is dominated by a vertex in V (T ) \L(T ). Proof. Every tree with more than two vertices does not have adjacent leaves. Then, for every vertex v ∈ L(T ), the only vertex adjacent to v in T is not in L(T ) and, by Proposition 4.30, dominates v.

Now that we know that we only need to consider dominated vertices, the major results can be presented. Theorem 4.32. Let G be a dually chordal graph and A ⊆ V (G) be a set of vertices, each being dominated by a vertex in V (G) \ A. Let G0 be the graph constructed from G by adding, for each v ∈ A, a vertex v∗ and the edge vv∗. Then, G0 is dually chordal. Moreover, there exists a tree T compatible with G such that L(T ) = A if and only if there exists a tree T 0 compatible with G0 such that L(T 0) = A∗ := {v∗, v ∈ A}. Proof. Let T be a tree compatible with G. Then, the tree T 0 such that V (T 0) = V (G) ∪ A∗ and E(T 0) = E(T ) ∪ {vv∗, v ∈ A} is compatible with G0, so this graph is dually chordal. Furthermore, if L(T ) = A, then L(T 0) = A∗. Conversely, suppose that there exists a tree T 0 compatible with G0 such that L(T 0) = A∗. 0 ∗ 0 0 Set T0 = T − A . Choose T so that |L(T0)| is maximized. Since C(G) ⊆ C(G ), every clique of 0 G induces a subtree of T and hence of T0. Then, T0 is compatible with G. Now we show that L(T0) = A, which will prove the claim. It is straightforward that L(T0) ⊆ A, otherwise any vertex in L(T0) \ A would also be a leave of T 0, which is a contradiction. Suppose that L(T0) 6= A. Take a vertex u ∈ A \L(T0) and let w be a vertex in V (G) \ A 0 dominating u and w be the vertex adjacent to u in T0hu, wi. 0 By Theorem 4.3, if, for each vertex x different from w and adjacent to u in T0, we add the edge wx to T0 and remove ux, we get a new tree T1 compatible with G such that the degree of w is bigger in T1 than in T0, u is a leaf of T1 and the remaining vertices have the same degree in T0 and T1. Then, L(T1) = L(T0) ∪ {u}, contradicting the way T0 was chosen. Therefore, L(T0) = A. Define the leafage of a chordal graph G as the minimum number of leaves of a clique tree of G. Similarly, if G is dually chordal, define the dual leafage of G as the minimum number of leaves of a compatible tree of G. Then, Theorem 4.33. Let G be a dually chordal graph and H be a basic chordal graph such that K(H) = G. Then, the dual leafage of G equals the leafage of H.

51 Proof. The compatible trees of G are exactly the clique trees of H. Therefore, both minima are equal.

Figure 4.8: A chordal graph whose leafage equals 2.

A polynomial time algorithm for ﬁnding the leafage of a chordal graph is known [15]. Note that a graph like H can also be obtained in polynomial time. For example, we can pick, for each edge uv of G, a clique containing {u, v}. The collection of those cliques in union with {{v} : v ∈ V (G)} gives a family F such that |F| ≤ |V (G)| + |E(G)|. In view of Theorem 4.15, L(F) is basic chordal and K(L(F)) = G. Then, we can set H equal to L(F). Therefore, the dual leafage of a dually chordal graph can be computed in polynomial time. Now we ﬁnd the relationship between the subject of this section and the concept of leafage.

Theorem 4.34. Let G, G0 and A be the same as in Theorem 4.32. Let also H0 be a basic chordal graph such that K(H0) = G0. Then, there exists a tree T compatible with G such that L(T ) = A if and only if the leafage of H0 equals |A|.

Proof. By Theorem 4.32 and the fact that A∗ is contained in the set of leaves of every compatible tree of G0, there exists a tree T compatible with G such that L(T ) = A if and only if the dual leafage of G0 equals |A|. The proof is complete by noting that, by Theorem 4.33, the dual leafage of G0 equals the leafage of H0.

The algorithm in [15] not only computes the leafage of a chordal graph, but also outputs a clique tree whose number of leaves equals the leafage of the graph. Let T 0 be the clique tree obtained when the algorithm is applied to H0. If the leafage of H0 equals |A|, then T 0 is a 0 0 ∗ 0 ∗ compatible tree of G such that L(T ) = A . Like in the proof of Theorem 4.32, let T0 = T −A . Thus, T0 is compatible with G and L(T0) ⊆ A. If L(T0) 6= A, then apply the procedure described at the end of the proof of Theorem 4.32 to increase the number of leaves until the set of leaves equals A.

52 Chapter 5

Detecting the families of clique trees of chordal graphs

So far, a few problems about clique trees, like finding one or calculating the leafage, have been considered. With regard to finding a single clique tree, we know from Theorem 4.2 that it can be done efficiently. If one wants to find all the clique trees of a chordal graph, it could happen that there are too many of them. Since the number of trees with a given set of vertices is exponential, so could be the case for the clique trees of a chordal graph. However, it is possible to find a pattern that allows the construction of all the clique trees and there are polynomial algorithms to count them [24]. The inverse problem had not apparently been studied and is of theoretical importance here. More clearly, take a family T of trees on the same vertex set V , the problem is to find a chordal graph G such that τ (G) = T , when possible. The current chapter shows that determining whether such a graph exists can be done by finding necessary and sufficient conditions that are testable in polynomial time with respect to |T | and |V |. The structural properties of clique trees and of the sets that induce subtrees in them are again exploited. It will also be possible to characterize all the chordal graphs whose family of clique trees equals T and a similar version of the problem, this time about compatible trees, will be studied.

5.1 A detection method which involves counting

In order to find the first necessary and sufficient conditions, it is required to develop a formula to count all clique trees of a chordal graph. The formula to be shown (Theorem 5.4) was already known. Anyway, it is useful to get a proof of it based on the concepts of the previous chapter. Recall that a relation R on a set A is an equivalence relation if it is reflexive, symmetric and transitive. A subset X of A is an equivalence class if the elements of X are pairwise related and X is maximal in this respect. The definitions of equivalence relation and equivalence class imply that different equivalence classes are disjoint and that every two elements a and b of A are related if only if they are in the same equivalence class. The quotient set of R, or A/R, is the set of all the equivalence classes of R. G Let G be a chordal graph and S ∈ S(G). RS is defined as the relation on CS such that C1 and C2 are related if and only if they intersect the same connected component of G − S. This is clearly an equivalence relation and there is one equivalence class per connected component of G − S that contains a vertex adjacent to all the vertices of S. The result we need about this relation is:

53 Proposition 5.1. Let G be a chordal graph and S ∈ S(G). Then, every equivalence class of G RS is in SC(G). G Proof. Let X be an equivalence class of RS and G[A] be the connected component of G − S that is intersected by the members of X. Let C be a clique of G[A∪S]. If C ∩A = ∅, then C ⊆ S. But S is not a maximal complete set of G[A ∪ S] because it is contained in any member of X. We infer from this contradiction that C ∩ A 6= ∅. Suppose that C is not a clique of G. Then, there exists a vertex v ∈ V (G) \ (A ∪ S) that is adjacent to all the vertices of C. Thus, v is adjacent to a vertex of A, which is a contradiction. Therefore, C is a clique of G. We conclude from the previous paragraph that C(G[A ∪ S]) ⊆ C(G). Now we prove that C(G[A ∪ S]) ∈ SC(G). G[A ∪ S] is a chordal graph. Then, we can pick a clique tree T1 of G[A ∪ S]. 0 Let CC be an edge of T1. We need to demonstrate that CC∩C0 , calculated with respect to G, is a subset of C(G[A ∪ S]). Suppose on the contrary that C00 is a clique of G such that 00 0 0 00 0 C ∈ CC∩C0 \C(G[A ∪ S]). Then, C ∩ C ⊆ C ∩ C ∩ C ⊆ S. By Theorem 4.5, C and C form a separating pair of G[A ∪ S]. Then, C \ C0 and C0 \ C are in diﬀerent connected components of G[A ∪ S] − C ∩ C0. However, G[A ∪ S] − C ∩ C0 is a connected graph because G[A] and G[S \ (C ∩ C0)] are connected and, if S \ (C ∩ C0) 6= ∅, there exists an edge of G such that one of its endpoints is in A and the other is in S \ (C ∩ C0). This gives a contradiction. Therefore, CC∩C0 ⊆ C(G[A ∪ S]). We can use this inclusion to get that C(G[A ∪ S]) = c S CC∩C0 . Consequently, C(G[A ∪ S]) ∈ SC(G). 0 CC ∈E(T1) Since X = CS ∩ C(G[A ∪ S]), X ∈ SC(G).

G Given a clique tree T of G, deﬁne now HT,S as the graph whose set of vertices is CS/RS , being X1 and X2 adjacent if and only if there exists an edge of T with one of its endpoints in X1 and the other endpoint in X2. Then: Proposition 5.2. Let G be a chordal graph, T be a clique tree of G and S ∈ S(G). Then, the graph HT,S is a tree.

G Proof. As CS and each equivalence class of RS induce a subtree of T , the number of edges of HT,S equals |V (HT,S)|−1. By the same reason, HT,S is connected. Therefore, HT,S is a tree. This result leads us to the definition of S-admissible edges. Let E0 be a set of edges of K(G) 0 G such that every e ∈ E has its endpoints in different equivalence classes of RS . Define HE0 as the G 0 graph such that V (HE0 ) = CS/RS and, for each e ∈ E , there exists an edge in HE0 connecting 0 the equivalence classes of the endpoints of e. Then, E is said to be S-admissible if HE0 is a tree. This definition allows to give another characterization of the clique trees of a chordal graph.

Theorem 5.3. Let G be a noncomplete chordal graph and T be a tree such that V (T ) = C(G). Let S(G) = {S1, ..., Sj}. The following are equivalent:

(a) T is a clique tree of G.

j S (b) E(T ) = Ei, being Ei Si-admissible for all 1 ≤ i ≤ j. i=1

54 G Figure 5.1: Suppose that G and S ∈ S(G) are such that CS/RS = {{C1,C2,C3}, {C4,C5}, {C6,C7}, {C8,C9}}. Then, the two sets of edges in the ﬁgure are S-admissible.

Proof. (a) ⇒ (b): See Proposition 5.2. (b) ⇒ (a): Let T0 be a clique tree of G and A1 be the set of all the edges of T0 such that the intersection of the endpoints of each of them equals S1. Define T1 = T0 − A1 + E1. It is not difficult to verify that T1 is a tree. Now we prove that T1 is a clique tree of G. Suppose that v ∈ V (G) is such that Cv does not induce a subtree of T1. This implies that the removal of some edge in A1 disconnects T0[Cv] and the endpoints of that edge are left in different connected components of T1[Cv]. As the intersection of those cliques equals S1 and v is in both of them, v ∈ S1.

On the other side, it is not diﬃcult to see that, by the construction, T1[CS1 ] is connected.

Since CS1 ⊆ Cv, all the cliques in CS1 are in the same connected component of T1[Cv], which is a contradiction. We conclude from this reasoning that T1 is a clique tree of G. Deﬁne recursively, for all 1 ≤ i ≤ j, Ai as the set of all the edges of Ti−1 such that the intersection of the endpoints of each of them equals Si and Ti = Ti−1 − Ai + Ei. By the same reasonings as before, Ti is a clique tree of G for all 1 ≤ i ≤ j. Furthermore, Tj = T . Therefore, T is a clique tree.

The formula for the number of clique trees of a chordal graph can now be found:

Theorem 5.4. Let G be a chordal graph. Then, the number of clique trees of G is equal to " !# Y Y |C /RG|−2 |X| S S |CS| . G S∈S(G) X∈ CS /RS Q Proof. By Theorem 5.3, it is clear that the total number of clique trees of G equals NS, S∈S(G) where NS is the number of possible S-admissible sets of edges, S ∈ S(G). We need to ﬁnd a formula for NS. G Let CS/RS = {X1, ..., Xk} and T be a clique tree of G. If we know that, for all 1 ≤ i ≤ k, the degree of Xi in HT,S is di, then, by Pr¨ufer’s proof of k − 2 Cayley’s formula [22], there are possible trees that HT,S can be. d1 − 1 ... dk − 1

55 If we now know what HT,S exactly is, then the number of possible S-admissible sets of edges k Q d that T can have is equal to |Xi| i . i=1 As the sum of the degrees of the vertices of HT,S equals 2k − 2,

" k # P k − 2 Y di NS = |Xi| = d1 − 1 ... dk − 1 i=1 d1 + ... + dk = 2k − 2 k ! " k # Y P k − 2 Y di−1 |Xi| |Xi| = d1 − 1 ... dk − 1 i=1 i=1 d1 + ... + dk = 2k − 2 k ! " k # k ! k !k−2 Y P k − 2 Y ei Y X |Xi| |Xi| = |Xi| |Xi| e1 ... ek i=1 i=1 i=1 i=1 e1 + ... + ek = k − 2 k ! Y k−2 = |Xi| |CS| i=1

If we want the sets, graphs and relations introduced in this chapter to be applied to obtain a method of detection of families of clique trees of chordal graphs, then we have to be able to identify them by looking at the clique trees and without looking at the graph. The following results indicate how that can be done. Given a family T of trees on the same vertex set V , there are two graphs associated with it that will be important for this and the next section. The graph HT has vertex set equal to T , 0 0 0 where T and T , T 6= T , are adjacent in HT if and only if there exist edges e and e such that T 0 = T − e + e0. Let E(T ) be the set of edges each of which is in at least one tree of T . Call ∗ ∗ 0 0 HT the graph such that V (HT ) = E(T ), in which e, e ∈ E(T ), e 6= e , are adjacent if and only 0 0 ∗ if there exists T ∈ T such that T − e + e ∈ T . For each connected component H of HT , we deﬁne P (H0) = S {u, v}. uv ∈ V (H0) Then we have:

Proposition 5.5. Let G be a chordal graph. Then, Hτ(G) is connected. Proof. Apply Theorem 4.2 and Proposition 4.7.

∗ Now we proceed to characterize the graph Hτ(G):

Proposition 5.6. Let G be a chordal graph and C1C2,C3C4 be two diﬀerent elements of ∗ E(τ (G)). Then, C1C2 and C3C4 are adjacent in Hτ(G) if and only if C1 ∩ C2 = C3 ∩ C4. ∗ Proof. Suppose that C1C2 and C3C4 are adjacent in Hτ(G). Let T be a clique tree of G such that T − C1C2 + C3C4 is also a clique tree. Then, C1,C2 ∈ T [C3,C4], so C3 ∩ C4 ⊆ C1 ∩ C2. Furthermore, by Theorem 4.2, |C1 ∩ C2| = |C3 ∩ C4|. Therefore, C1 ∩ C2 = C3 ∩ C4. Conversely, suppose that C1 ∩C2 = C3 ∩C4. Let S = C1 ∩C2 and G[A1],G[A2],G[A3],G[A4] be the connected components of G−S intersecting C1,C2,C3,C4, respectively. Also let B be the set of the vertices of the other connected components, if any. Then, by Theorem 4.5, A1 6= A2 and A3 6= A4. We consider three cases:

56 ∗ Figure 5.2: A family T of trees and the graphs HT and HT . Each edge of HT is labeled with the two elements of E(T ) that the trees must exchange to get one from the other. Each edge 0 ∗ 0 ee of HT is labeled with all the pairs of trees that can exchange e and e to get one from the other.

1) A1,A2,A3,A4 are all diﬀerent: Let T1 be a clique tree of G[A1 ∪ B ∪ S] and Ti, i = 2, 3, 4, be a clique tree of G[Ai ∪ S]. Let T = T3 + C1C3 + T1 + C1C2 + T2 + C2C4 + T4. Now we prove that T is a clique tree. Let v ∈ V (G). If v 6∈ S, then v is in A1 ∪ B, A2, A3 or A4. If v ∈ A1 ∪ B, then T [Cv] = T1[Cv], which is a subtree. If v ∈ Ai, i = 2, 3, 4, then T [Cv] = Ti[Cv], also a subtree. If v ∈ S, then T [Cv] is formed by the subtrees T1[Cv ∩C(G[A1 ∪B ∪S])] and Ti[Cv ∩C(G[Ai ∪S])], i = 2, 3, 4, all joined together by the edges C1C3, C1C2 and C2C4. Therefore, T [Cv] is a subtree. We conclude that T is a clique tree. Similarly, T − C1C2 + C3C4 is also a clique tree. Therefore, ∗ C1C2 and C3C4 are adjacent in Hτ(G). 2) Two of the sets are equal: Suppose without loss of generality that A1 = A3. Let T = T1 +C1C2 +T2 +C2C4 +T4. Then, T is a clique tree of G and so is T −C1C2 +C3C4. Therefore, ∗ C1C2 and C3C4 are adjacent in Hτ(G). 3) There are two couples of equal sets: Suppose without loss of generality that A1 = A3 and A2 = A4. Let T = T1 + C1C2 + T2. Then, T is a clique tree of G and so is T − C1C2 + C3C4. ∗ Therefore, C1C2 and C3C4 are adjacent in Hτ(G). In combination with Theorems 4.4 and 4.5, we have:

∗ Corollary 5.7. Let G be a chordal graph. Then, the number of connected components of Hτ(G) equals |S(G)| and each of them is a complete subgraph.

Now we proceed to study those connected components.

Proposition 5.8. Let G be a chordal graph, C1C2 ∈ E(τ (G)) and C3 be another clique such ∗ that C1 ∩ C2 ⊆ C3. Then, C1C2 is adjacent to C1C3 or to C2C3 in Hτ(G).

Proof. Let T be a clique tree of G such that C1C2 ∈ E(T ). Then, C1 ∈ T [C2,C3] or C2 ∈ T [C1,C3]. In the ﬁrst case, T − C1C2 + C2C3 is a clique tree, so C1C2 and C2C3 are adjacent in ∗ Hτ(G). In the second case, T − C1C2 + C1C3 is a clique tree. Thus, C1C2 and C1C3 are adjacent ∗ in Hτ(G).

57 0 Proposition 5.9. Let G be a chordal graph, C1C2 ∈ E(τ (G)), C1 ∩ C2 = S, and H be the ∗ 0 connected component of Hτ(G) containing C1C2. Then, P (H ) = CS. Proof. Let C ∈ P (H0). Take C0 such that CC0 ∈ V (H0). Then, by Proposition 5.6 and Corollary 0 0 5.7, C ∩ C = C1 ∩ C2 = S and hence C ∈ CS. Therefore, P (H ) ⊆ CS. 0 Conversely, let C ∈ CS. If C = C1 or C = C2, then clearly C ∈ P (H ). Otherwise, by ∗ ∗ Proposition 5.8, C1C2 is adjacent to CC1 in Hτ(G) or C1C2 is adjacent to CC2 in Hτ(G). Then, 0 0 0 CC1 ∈ V (H ) or CC2 ∈ V (H ). In either case, we conclude that C ∈ P (H ). It follows that 0 CS ⊆ P (H ). 0 Therefore, P (H ) = CS.

Now that the set CS could be identiﬁed by looking at the clique trees, we do the same with G the relation RS . G Proposition 5.10. Let G be a chordal graph, S ∈ S(G) and C1,C2 ∈ CS. Then, (C1,C2) ∈ RS 0 0 ∗ 0 if and only if C1C2 6∈ V (H ), where H is the connected component of Hτ(G) such that P (H ) = CS.

G 0 Proof. Suppose that (C1,C2) ∈ RS . If C1 ∩ C2 6= S, then C1C2 cannot be a vertex of H . If C1 ∩ C2 = S, then C1 and C2 do not form a separating pair. Thus, C1C2 is the edge of no ∗ 0 clique tree of G and cannot be a vertex of Hτ(G), let alone of H . G Now suppose that (C1,C2) 6∈ RS . Then, C1 and C2 form a separating pair and C1 ∩ C2 = S. 0 By the former, C1C2 ∈ E(τ (G)) and, by the latter, C1C2 ∈ V (H ).

All the previous results are enough to state the first necessary and sufficient conditions for a family of trees to be the family of clique trees of a chordal graph. 0 ∗ T Let T be a family of trees and H be a connected component of HT . Define RH0 as the 0 T 0 relation on P (H ) such that (u, v) ∈ RH0 if and only if uv 6∈ V (H ). Then: Theorem 5.11. Let T be a family of trees with common vertex set V and Q be the family of ∗ connected components of HT . Then, there exists a chordal graph G such that τ (G) = T if and only if the following three conditions hold:

0 ∗ T 1. For every connected component H of HT , RH0 is an equivalence relation. 0 ∗ 0 2. For every connected component H of HT and T ∈ T , P (H ) and each equivalence class T of RH0 induce a subtree of T . " !# Y Y 0 |P (H0)/RT |−2 3. |T | = |X| |P (H )| H0 0 H ∈Q X∈ P (H0)/RT H0

Proof. Suppose that T is the family of clique trees of a graph G. Then, condition 1 is a consequence of Proposition 5.10, condition 2 is a consequence of Propositions 5.9 and 5.1 and condition 3 is a consequence of the propositions mentioned previously and Theorem 5.4. S 0 T S Conversely, suppose that the three conditions are satisﬁed. Let F = ( P (H )/RH0 ) H0∈Q {P (H0): H0 ∈ Q} S {{v} : v ∈ V } and G = L(F). Then, by Theorem 1.4 and condition 2, G is a chordal graph. By Proposition 4.14, |C(G)| = |V | and every T ∈ T corresponds to a clique tree of G.

58 0 ∗ Let T be a clique tree of G and H be a connected component of HT . By the construction 0 T 0 T of G, the graph whose vertex set is P (H )/RH0 and such that X1 and X2 in P (H )/RH0 are adjacent if and only if T has an edge with one endpoint in X1 and the other endpoint in X2 is a tree. Note that an edge like the one mentioned in the previous sentence must be in V (H0). Conversely, by condition 1, every e ∈ V (H0) has endpoints in diﬀerent equivalence classes of T RH0 . Then, reasoning like in the proof of Theorem 5.4, the number of possible ways in which the edges that are in V (H0) can be chosen in the procedure of constructing a clique tree of G is at Q Y |X| ! |P (H0)/RT |−2 |X| 0 H0 most 0 T |P (H )| . X∈ P (H )/R 0 X∈ P (H0)/RHT H0 After making this choice for each H0 ∈ Q, no more edges should be chosen to obtain a clique tree of G. Otherwise, the resulting clique tree would have more edges than any T ∈ T , which is also a clique tree of G, a contradiction. " !# Y Y |P (H0)/RT |−2 |X| 0 H0 Therefore, |τ (G)| ≤ |P (H )| . 0 H ∈Q X∈ P (H0)/RT H0 By condition 3, |τ (G)| ≤ |T |. We had also found that T ⊆ τ (G). Then, τ (G) = T .

∗ As an example, consider the family of trees of Figure 5.2. In this case, HT has four con- 0 0 nected components. Let H1 be the connected component with vertices 12 and 13, and H2 0 0 be the connected component with vertices 34 and 45. Then, P (H1) = {1, 2, 3}, P (H2) = T T {3, 4, 5}, R 0 = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} and R 0 = {(3, 3), (4, 4), (5, 5), (3, 5), (5, 3)}. H1 H2 0 T Thus, these relations are equivalence relations, in a way that P (H1)/R 0 = {{1, 3}, {2}} and H1 0 T P (H2)/R 0 = {{3, 5}, {4}}. This allows us to see that condition 2 is satisﬁed. Furthermore, H2

Y Y 0 |P (H0)/RT |−2 [( |X| ) |P (H )| H0 ] = 0 H ∈Q X∈ P (H0)/RT H0

|P (H0 )/RT |−2 |P (H0 )/RT |−2 0 1 H0 0 2 H0 (|{1, 3}|.|{2}|) . |P (H1)| 1 . (|{3, 5}|.|{4}|) . |P (H2)| 2 =

(2 . 1) . 32−2 . (2 . 1) . 32−2 = 2 . 2 = 4 We have not considered the other connected components because it is trivial to check that they satisfy conditions 1 and 2 and that they do not alter the product. As the result is equal to the amount of trees of the family, condition 3 is also satisﬁed. Therefore, the trees of Figure 5.2 form a family of clique trees of some chordal graph.

5.2 Another detection method

In this section, new necessary and suﬃcient conditions for a family of trees to be a family of clique trees of a chordal graph are derived. These conditions are purely structural and will not involve counting. Rather, they are based on the idea of building a graph that can potentially have the given trees as clique trees. Once again, it will be useful to ﬁnd new characterizations of the clique trees of a chordal graph and to identify the subtree inducing sets by only looking at the trees. Those are the things that are done in the following lines.

59 Proposition 5.12. Let G be a chordal graph and B be a generating subfamily of SC(G). Then: (a) T is a clique tree of G if and only if every member of B induces a subtree of T .

(b) Let F = B S {{C} : C ∈ C(G)}. Then, the intersection graph of F is chordal and has the same clique trees as G. Proof. (a) Let T be a clique tree of G. Then, since B ⊆ SC(G), every member of B induces a subtree of T . Conversely, suppose that T is a tree with vertex set C(G) such that every member of B induces a subtree of T . For every v ∈ V (G), either Cv is a unit set or it can be expressed as the connected union of members of B. Thus, Cv induces a subtree of T . Therefore, T is a clique tree. (b) Set G0 = L(F). We can represent G0 as the intersection graph of subtrees of any clique tree of G, so it is a chordal graph. By Proposition 4.14, the family of cliques of L(F) consists of all the sets DC , C ∈ C(G). For ∼ F ∈ F, the set of cliques of L(F) containing F is {DC : C ∈ F }. Therefore, {Cv}v∈V (G0) = F. By the construction of F, it is a consequence of part (a) that T is a clique tree of G if and only if every member of F induces a subtree of T , that is, if and only if T is a clique tree of G0. Therefore, G and G0 have the same clique trees.

In our context, Proposition 5.12 means that if someone found a chordal graph making the answer to our decision problem aﬃrmative, not revealing what the graph is but revealing one family generating all the subsets that induce subtrees of every clique tree of the graph, then we would be able to verify it ourselves by constructing another graph. Consequently, knowing the generating family is almost as important as knowing the chordal graph itself. This also suggests that, given T , trying to derive from it a generating family for a potential chordal graph with family of clique trees equal to T might be a very useful approach. We know how to do this by using Proposition 5.9. We ﬁnd below an alternative way that could be simpler.

Proposition 5.13. Let G be a chordal graph, S ∈ S(G) and C1C2 ∈ E(τ (G)) be such that S C1 ∩ C2 = S. Deﬁne TG[C1,C2] = T [C1,C2]. Then, TG[C1,C2] = CS. T ∈ τ(G)

Proof. Let C be any element of TG[C1,C2] and T ∈τ (G) be such that C ∈ T [C1,C2]. Since T is a clique tree, C1 ∩ C2 ⊆ C and hence C ∈ CS. Therefore, TG[C1,C2] ⊆ CS. Now suppose that C ∈ CS. If C = C1 or C = C2, then it is clear that C ∈ TG[C1,C2]. ∗ Otherwise, by Proposition 5.8, C1C2 is adjacent to CC1 in Hτ(G) or C1C2 is adjacent to CC2 in ∗ Hτ(G). Suppose without loss of generality that the ﬁrst is true. Let T be a clique tree of G such that T − CC1 + C1C2 is also a clique tree. Then, C ∈ T [C1,C2] and hence C ∈ TG[C1,C2]. Therefore, CS ⊆ TG[C1,C2] and the equality follows. With the help of Propositions 5.12 and 5.13, we obtain:

Theorem 5.14. Let T be a family of trees, all having the same vertex set V , T1 ∈ T , F = 0 S {T [u, v]: uv ∈ E(T1)} and F = F {{v} : v ∈ V }. Then, there exists a chordal graph G such that τ (G) = T if and only if L(F 0) is chordal and τ (L(F 0)) = T . Proof. Suppose that there exists a chordal graph G such that τ (G) = T . Then, by Proposition 5.13 and Theorem 4.13, F is a generating subfamily of SC(G) and, by Proposition 5.12, L(F 0) is chordal and τ (L(F 0)) = T .

60 The converse is clearly true. We just need to set G = L(F 0).

In view of Theorem 5.14, the answer to our problem solely depends on whether L(F 0) is a solution or not. In order to be a solution, two natural conditions arise, namely, all the members of T must be clique trees of L(F 0) and no other tree can be a clique tree of L(F 0). If L(F 0) is a solution to the problem and we want to apply the procedure described in Theorem 4.2 to L(F 0) to ﬁnd clique trees, then we need to know what the family of cliques of 0 0 L(F ) is. By Proposition 4.14, the family of cliques of L(F ) consists of all the sets Dv = {F ∈ F 0 : v ∈ F }, v ∈ V . In order to give weights to the edges of the clique graph, we note that |Du ∩ Dv| = |{F ∈ F : {u, v} ⊆ F }|. Additionally, if T and T 0 are two trees in T such that T 0 = T − wx + uv, we must have that Du ∩ Dv = Dw ∩ Dx. This motivates the following result:

Theorem 5.15. Let T be a family of trees, all having the same vertex set V , T1 ∈ T and F = {T [u, v]: uv ∈ E(T1)}. For each pair u, v ∈ V , u 6= v, deﬁne Duv = {F ∈ F : {u, v} ⊆ F }. Then, there is a chordal graph G such that τ(G) = T if and only if the following conditions are satisﬁed:

1. For all F ∈ F and T ∈ T , T [F ] is a subtree of T .

2. For all u, v ∈ V , u 6= v, T ∈ T and wx ∈ E(T ) such that {w, x} ⊆ T [u, v] and Duv = Dwx, T − wx + uv ∈ T .

Proof. Suppose that there is a chordal graph G such that τ (G) = T . Define F 0 like in Theorem 5.14. Then, τ (L(F 0)) = T . As the dual of the clique family of L(F 0) is isomorphic to F 0 (see proof of Proposition 5.12), condition 1. is satisfied. Let T be any tree in T and suppose that w and x are two vertices adjacent in T and contained in T [u, v] such that Duv = Dwx. Then, by Theorem 4.2 and the remark previous to this theorem, T + uv − wx is a clique tree of L(F 0), that is, T + uv − wx ∈ T . Conversely, suppose that conditions 1. and 2. are satisfied. Then, by condition 1., L(F 0) is 0 0 a chordal graph such that T ⊆ τ (L(F )). Now, let T ∈ T and T be adjacent to T in Hτ (L(F 0)). 0 Take the edges uv and wx such that T = T − wx + uv. Then, Duv = Dwx and, by condition 2., 0 T ∈ T . From this reasoning and the fact that, by Proposition 5.5, Hτ (L(F 0)) is connected, we conclude that T = τ (L(F 0)).

It is clear that F and the sets Duv can be found in polynomial time with respect to |T | and |V |. Condition 1. can also be tested in polynomial time. Moreover, for each pair of diﬀerent vertices u, v and T ∈ T , the number of edges wx in T such that Duv = Dwx cannot be larger than |V | − 1. Therefore, the number of operations necessary to test condition 2. is polynomial. As a conclusion, the whole problem can be solved polynomially. We could also have arrived at this conclusion by using the results of the previous section.

Let us now discuss some examples. Consider the trees of Figure 5.3. For them, F = {{1, 3}, {2, 3}, {3, 4}, {2, 3, 4, 5}}. This family clearly satisfies the condition 1 of Theorem 5.15. However, D25 = D35 and the tree obtained by removing the edge 25 from the first tree of Figure 5.3 and adding 35 to it is not in the family. Therefore, this is not a family of clique trees of a chordal graph because condition 2 is not satisfied. Now we offer an example where condition 2 is satisfied but condition 1 is not. Let T be the family of trees in Figure 5.4. Then, F = {{1, 2, 3}, {2, 3, 4}, {1, 2, 3, 4}}. The only equalities

61 Figure 5.3: A family of trees which satisﬁes condition 1 of Theorem 5.15, but that does not satisfy condition 2.

Figure 5.4: A family of trees satisfying condition 2 of Theorem 5.15 but not satisfying condition 1.

between sets Duv are given by D12 = D13 and D24 = D34. The fact that T1 −12+13, T1 −34+24, T3 − 13 + 12, T3 − 24 + 34, T4 − 12 + 13, T4 − 24 + 34, T5 − 13 + 12 and T5 − 34 + 24 are all in T means that condition 2 is satisfied. However, T2[{1, 2, 3}] and T2[{2, 3, 4}] are not subtrees of T2. Therefore, the condition 1 is not satisfied. Hence, T is not a family of clique trees of a chordal graph, either. Finally, let T now be the family of trees in Figure 5.5, which had already been considered in Figure 5.2. We know from the previous section that this is a family of clique trees of some chordal graph. We leave it to the reader to verify that the conditions 1 and 2 are satisfied. F = {{1, 3}, {3, 5}, {1, 2, 3}, {3, 4, 5}}, and the graph L(F 0) appears in the lower part of Figure 5.5. It is easy to use Theorem 4.2 to check that T = τ (L(F 0)).

5.3 Finding all chordal graphs with a given family of clique trees

In this section, we use the knowledge from the previous one to ﬁnd, given a family T of trees with common vertex set V , a general formula for all the graphs that have T as a family of clique trees. As a ﬁrst step, given a chordal graph G, we characterize the graphs with the same clique trees as G.

62 Figure 5.5: A family of trees for which the answer to the problem is aﬃrmative and the chordal graph, given by Theorem 5.14, having them as all its clique trees. Its weighted clique graph appears at the lower right. Every maximum weight spanning tree of it must have the edges 13 and 35. In order to complete it, we must choose from 12 and 23 and from 34 and 45. This conﬁrms that the family of clique trees of the chordal graph is composed of just the trees above.

Every graph can be determined by its dual clique family, the ﬁrst being the intersection graph of the second. For the case of G, the sets CS, S ∈ S(G), can be expressed as the intersection of members of the dual clique family. If G0 is another chordal graph with the same clique trees 0 as G, then the sets of the form CS, S ∈ S(G ), are the same as in G because, as we saw in Proposition 5.13, there is an expression for them in terms of the clique trees. Again, these sets can be expressed as the intersection of members of the dual clique family of G0. These ideas and some others lead to the following theorem:

Theorem 5.16. Let G and G0 be two chordal graphs. Then, G and G0 have the same clique trees if and only if G0 = L(F), where F is a separating subfamily of SC(G) such that, for every T S ∈ S(G), F = CS. F ∈F, CS ⊆F Proof. Suppose that G0 = L(F), where F satisﬁes the conditions of the statement of the theorem. If we repeat the reasoning of Proposition 5.12, part (b), then we also get that DC(G0) =∼ F. Therefore, since F ⊆ SC(G), every clique tree of G is a clique tree of G0. Now, let T be a clique tree of G0. Then, every member of F induces a subtree of T . The T condition that, for all S ∈ S(G), F = CS, implies that CS induces a subtree of T . F ∈F, CS ⊆F Therefore, by Proposition 5.12 and Theorem 4.13, T is a clique tree of G. It follows that G and G0 have the same clique trees. Conversely, suppose that G and G0 are two chordal graphs with the same clique trees. Then, SC(G) = SC(G0). Set F = DC(G0). Thus, G0 =∼ L(F) and, by the previous statement, F ⊆ SC(G). Now, let S ∈ S(G) and C1C2 ∈ E(τ (G)) such that C1 ∩ C2 = S. The equality

63 of clique trees for both graphs implies that TG[C1,C2] = TG0 [C1,C2]. By Proposition 5.13, TG[C1,C2] = CS. On the other hand, also by Proposition 5.13, TG0 [C1,C2] can be expressed as an intersection of members of the dual clique family of G0, that is, as an intersection of members T of F . The equality F = CS immediately follows. F ∈F, CS ⊆F

We know that when, given T , the question whether there is a chordal graph whose family of clique trees equals T has an aﬃrmative answer, we can use Theorem 5.14 to construct a graph with the required clique trees. By combining this with Theorem 5.16, we will be able to characterize all the chordal graphs with family of clique trees equal to T . 0 Let T , T1, F and F be the same as in Theorem 5.14, i.e., T a family of trees on the same 0 set V of vertices, T1 ∈ T , F = {T [u, v]: uv ∈ E(T1)} and F = F ∪ {{v} : v ∈ V }. Deﬁne the span of F, Sp(F), as the family of unit sets contained in V plus all the sets that can be obtained as connected unions of members of F. Then we have: Theorem 5.17. Let Ch(T ) = {G : τ (G) = T}. Then, one of the following is true: ∗ Ch(T ) = ∅. ∗ G ∈ Ch(T ) if and only if G = L(F 00), where F 00 is a separating subfamily of Sp(F) such that, T for all uv ∈ E(T1), F = T [u, v]. F ∈F00, {u,v}⊆F Proof. Suppose that Ch(T ) 6= ∅. Then, by Theorem 5.14, L(F 0) ∈ Ch(T ) and, by Proposition 0 5.13 and Theorem 4.13, Sp(F) = SC(L(F )). Let uv ∈ E(T1) and F ∈ Sp(F) be such that {u, v} ⊆ F . By the above, F induces a subtree of every T ∈ T . Thus, T [u, v] ⊆ F for every T ∈ T , and hence T [u, v] ⊆ F . Therefore, for every F ∈ Sp(F), {u, v} ⊆ F if an only if T [u, v] ⊆ F . We conclude by applying Theorem 5.16 to L(F 0).

As an example, consider the family T in Figure 5.6. It holds that Ch(T ) 6= ∅, F = {{1, 3}, {2, 3}, {3, 4}, {2, 3, 4, 5}}, and L(F 0) equals the graph G in the figure. The figure also displays another graph G0 ∈ Ch(T ). G0 can be viewed as the intersection graph of the family F 00 = {{1}, {2}, {4}, {5}, {1, 2, 3}, {1, 3, 4}, {2, 3, 4, 5}}. It is not hard to check that F 00 ⊆ Sp(F). Let T1 be the tree in the upper left of the figure. The members of F 00 that contain {1, 3} are {1, 2, 3} and {1, 3, 4}. Their intersection equals T [1, 3]. The members of F 00 that contain {2, 3} are {1, 2, 3} and {2, 3, 4, 5}. Their intersection equals T [2, 3]. The members of F 00 that contain {3, 4} are {1, 3, 4} and {2, 3, 4, 5}. Their intersection equals T [3, 4]. The only member of F 00 that contains {2, 5} is {2, 3, 4, 5}, which is equal to T [2, 5]. Therefore, we see that this example is in agreement with Theorem 5.17.

5.4 Detecting the families of compatible trees of dually chordal graphs

The subject of this section is not much diﬀerent from the subject of the previous ones. But our focus will be now on dually chordal graphs. More precisely, given a family T of trees with

64 Figure 5.6: A family of trees and two graphs having them as their clique trees.

common set of vertices V , it has to be determined whether there exists a dually chordal graph whose family of compatible trees is T . This section will not be as detailed as the others, but is intended to be an overview. It is interesting to note that a family of compatible trees of a dually chordal graph is also a family of clique trees of a chordal graph. What is more, we can say the following:

Proposition 5.18. Let T be a family of trees with common vertex set V . Then, T is the family of compatible trees of a dually chordal graph if and only if T is the family of clique trees of a basic chordal graph.

Proof. Use the deﬁnition of basic chordal graph and Theorem 4.15.

Nevertheless, not every family of clique trees of a chordal graph is the family of compatible trees of a dually chordal graph. Consider the trees in Figure 5.6. They are the clique trees of the graphs below them. However, let us see that there is no dually chordal graph having them as the family of its compatible trees. If such a graph existed, then 13 would be one of its edges and, by the deﬁnition of compatible tree, {2, 3, 4, 5} would be a complete set. But the graph should have more edges; otherwise, the path 13245 would be a compatible tree. Suppose that one additional edge is 12. Then, 2 is a universal vertex, and hence the star centered at 2 is a compatible tree, which is a contradiction. The same contradiction will arise if any other edge is added. Therefore, no dually chordal graph has the trees of Figure 5.6 as its only compatible trees.

In view of Proposition 5.18, we can restrict the analysis of the problem of this section to families of clique trees of chordal graphs. Let T be the family of clique trees of a chordal graph and T ∈ T . Let F = {T [a, b]: a, b ∈ V (T )}. It is not diﬃcult to prove that F is a family of subtrees of every tree in T .

65 If there exists a dually chordal graph G such that the family of compatible trees of G is T , consider F 0 = {T [a, b]: ab ∈ E(G)}. By using the properties of compatible trees, we infer that the members of F 0 are complete sets of G. Consequently, G = S(F 0). Moreover, a technique similar to that of the proof of Proposition 5.13 allows to prove that, if u, v are adjacent in T , then T [u, v] = T N[w] = {w ∈ V (G): N[u] ∩ N[v] ⊆ N[w]}. w∈N[u]∩N[v] Conversely, suppose that there exists F 0 ⊆ F such that S(F 0) = G and, for all uv ∈ E(T ), T NG[w] = T [u, v]. By the construction of G, T is compatible with it. Consider a w∈NG[u]∩NG[v] chordal graph G0 whose family of clique trees is T . The equality in this paragraph, combined with Propositions 4.19 and 5.13 and Theorems 4.13 and 4.18, implies that the basis of SDC(G) is equal to the basis of SC(G0). Therefore, by Theorem 4.11, the clique trees of G0 are exactly the compatible trees of G, that is, the family of compatible trees of G is T . As a consequence, we have the following theorem:

Theorem 5.19. Let T be the family of clique trees of a chordal graph, T ∈ T and F = {T [a, b]: a, b ∈ V (T )}. Then, T is the family of compatible trees of a dually chordal graph G if and only if G = S(F 0), where F 0 is a subfamily of F, and, for all uv ∈ E(T ), the set of vertices w such that NG[u] ∩ NG[v] ⊆ NG[w] equals T [u, v]. Theorem 5.19 makes our problem one about set theory. Let us illustrate it with a couple of examples. Consider the trees in Figure 5.5. They are the clique trees of the chordal graph also shown there. The basis for this chordal graph is composed of {1, 2, 3}, {1, 3}, {3, 4, 5} and {3, 5}. However, it is not diﬃcult to see that the two section of these sets is a dually chordal graph with more than four compatible trees. But if we take T [1, 2] = {1, 2, 3}, T [1, 5] = {1, 3, 5} and T [3, 4] = {3, 4, 5}, then we get a family F 0 in the conditions of Theorem 5.19. Therefore, the trees in Figure 5.5 are all the compatible trees of a dually chordal graph (see Figure 5.7).

Figure 5.7: A dually chordal graph and a valuation for its edges, according to Theorem 4.3, that allows to see that its compatible trees are those in Figure 5.5.

Consider again the trees in Figure 5.6 and let T be the tree at the left. Then, T [1, 2] = {1, 2, 3}, T [1, 3] = {1, 3}, T [1, 4] = {1, 3, 4}T [1, 5] = {1, 2, 3, 4, 5}, T [2, 3] = {2, 3}, T [2, 4] = {2, 3, 4}, T [2, 5] = {2, 3, 4, 5}, T [3, 4] = {3, 4} and T [3, 5] = {2, 3, 4, 5}. Suppose that there exists a family F 0 satisfying the conditions of Theorem 5.19. It is easy to 0 0 see that T [1, 5] should not be in F . Furthermore, T [2, 5] should be in F , otherwise NS(F 0)[2] ∩ 0 0 NS(F 0)[5] = ∅. Since T [2, 3] = {2, 3}, we need that T [1, 2] ∈ F and that T [1, 4] 6∈ F so that NS(F 0)[2] ∩ NS(F 0)[3] * NS(F 0)[4]. However, if we now take into account that T [3, 4] = {3, 4}, the conclusion this time is that T [1, 4] ∈ F 0 and that T [1, 2] 6∈ F 0, which is a contradiction.

66 We have seen again that the family of trees of Figure 5.6 is not the family of compatible trees of a dually chordal graph.

Theorem 5.19 could be used satisfactorily because the families considered were small. How- ever, it could be quite complicated for more general instances. The complexity of determining whether a family of trees is the family of compatible trees of a dually chordal graph is left as an open question.

67 Chapter 6

Conclusions

A list of the main contributions of this work is given below:

• The proofs of how, despite their different definitions, vertices with a maximum neighbor, simplicial, power simplicial, doubly simplicial and simple vertices lead to analogous properties of chordal, dually chordal, power chordal, doubly chordal and strongly chordal graphs. • A better understanding of the characterizations of dually chordal graphs through a thor- ough study of compatible trees. • The introduction of basic chordal graphs and a characterization for them that makes their identification easy enough. • Showing, given a dually chordal graph G, how the basic chordal graphs whose clique trees are exactly the compatible trees of G can be found. It follows from this that every problem about the compatible trees of a dually chordal graph can be reduced to a problem about the clique trees of a chordal graph. This is a relevant fact because several problems about the clique trees of a chordal graph, like the leafage of a chordal graph, have been studied and efficiently solved. • Showing how a couple of problems about trees can be solved efficiently in polynomial time. Those problems are: determining whether a set of vertices is the set of leaves of some compatible tree of a given dually chordal graph; and, given a family of trees, deciding whether it is the family of clique trees of some chordal graph.

If we continue thinking about these questions, then many other related problems easily arise and are left open. As an example, three of them are given below:

• Given a dually chordal graph G and a family F whose elements are the vertices of G, the following question is asked: What are the necessary and suﬃcient conditions that F must satisfy so that the family of compatible trees of G consists of all the spanning trees T such that every member of F induces a subtree of T ? • Two subclasses of chordal graphs, namely DV and RDV graphs, can be characterized by the existence of special types of clique trees and have dual classes that in turn can be characterized by the existence of special types of compatible trees. Therefore, basic DV/RDV graphs can be deﬁned similarly to basic chordal graphs. What properties do these new classes have and how can they be characterized?

68 • Finally, as already mentioned in Chapter 5, it is interesting in the context of this work to ﬁnd the algorithmic complexity of determining, given a family T of trees, whether T is the family of compatible trees of some dually chordal graph.

Finding an answer for these and other questions is part of the goals set for the author for the short and mid-term.

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