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A Joint Study of Chordal and Dually Chordal Graphs

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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 II 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 sim- plicial 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.
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