Pablo De Caria

On the correspondence between tree representations of chordal and dually chordal graphs

Chordal graphs and their clique graphs (called dually chordal graphs) possess characteristic tree representations, namely, the clique tree and the compatible tree, respectively. The following problem is studied: given a chordal graph G, determine if the clique trees of G are exactly the compatible trees of the clique graph of G. This leads to a new subclass of chordal graphs, basic chordal graphs, which is here characterized. The question is also approached backwards: given a dually chordal graph G, we find all the basic chordal graphs with clique graph equal to G. This approach leads to the possibility of considering several properties of clique trees of chordal graphs and finding their counterparts in compatible trees of dually chordal graphs.

On minimal vertex separators of dually chordal graphs: Properties and characterizations

Many works related to dually chordal graphs, their cliques and neighborhoods were published by Brandstadt et al. (1998) [1] and Gutierrez (1996) [6]. We will undertake a similar study by considering minimal vertex separators and their properties instead. We find a necessary and sufficient condition for every minimal vertex separator to be contained in the closed neighborhood of a vertex and two major characterizations of dually chordal graphs are proved. The first states that a graph is dually chordal if and only if it possesses a spanning tree such that every minimal vertex separator induces a subtree. The second says that a graph is dually chordal if and only if the family of minimal vertex separators is Helly, its intersection graph is chordal and each of its members induces a connected subgraph. We also found adaptations for them, requiring just O(|E(G)|) minimal vertex separators if they are conveniently chosen. We obtain at the end a proof of a known characterization of the class of hereditary dually chordal graphs that relies on the properties of minimal vertex separators.

Comparing trees characteristic to chordal and dually chordal graphs

Abstract Chordal and dually chordal graphs possess characteristic tree representations, namely, clique trees and compatible trees, respectively. The following problem is studied: given a chordal graph G, it has to be determined if the clique trees of G are exactly the compatible trees of K ( G ) . This does not always happen. A necessary and sufficient condition so that it is true, in terms of the minimal vertex separators of the graph, is found.

On basic chordal graphs and some of its subclasses

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph.In this work, we consider some subclasses of basic chordal graphs. One of them is the class of hereditary basic chordal graphs, which will turn out to have many possible characterizations. Those characterizations will show that the class was already studied, but under different names and in different contexts.We also study the connection between basic chordal graphs and some subclasses of chordal graphs with special clique trees, like D V graphs and R D V graphs. As a result, it will be possible to define the classes of basic D V graphs and basic R D V graphs.Additionally, we study the behavior of the clique operator over all the considered subclasses.

The Images of the Clique Operator and Its Square are Different

Let G be the class of all graphs and K be the clique operator. The validity of the equality KG=K2G has been an open question for several years. A graph in KG but not in K2G is exhibited here.

Determining what sets of trees can be the clique trees of a chordal graph

Chordal graphs have characteristic tree representations, the clique trees. The problems of finding one or enumerating them have already been solved in a satisfactory way. In this paper, the following related problem is studied: given a family of trees, all having the same vertex set V, determine whether there exists a chordal graph whose set of clique trees equals . For that purpose, we undertake a study of the structural properties, some already known and some new, of the clique trees of a chordal graph and the characteristics of the sets that induce subtrees of every clique tree. Some necessary and sufficient conditions and examples of how they can be applied are found, eventually establishing that a positive or negative answer to the problem can be obtained in polynomial time. If affirmative, a graph whose set of clique trees equals is also obtained. Finally, all the chordal graphs with set of clique trees equal to are characterized.

Minimal vertex separators and new characterizations for dually chordal graphs

Abstract Many works related to dually chordal graphs, their cliques and neighborhoods were published. We will undertake a similar study but by considering minimal separators and their properties. Moreover we find new characterizations of dually chordal graphs.

On second iterated clique graphs that are also third iterated clique graphs

Abstract Iterated clique graphs arise when the clique operator is applied to a graph more than once. Determining whether a graph is a clique graph or an iterated clique graph is usually a difficult task. The fact that being a clique graph and being an iterated clique graph are not equivalent things has been proved recently. However, it is still unknown whether the classes of second iterated clique graphs and third iterated clique graphs are the same. In this work we find classes of graphs, defined by means of conditions on the clique size and the structure of the clique intersections, whose second iterated clique graphs are also third iterated clique graphs.

Special eccentric vertices for the class of chordal graphs and related classes

A vertex is simplicial if the vertices of its neighborhood are pairwise adjacent. It is known that, for every vertex v of a chordal graph, there exists a simplicial vertex among the vertices at maximum distance from v. Here we prove similar properties in other classes of graphs related to that of chordal graphs. Those properties will not be in terms of simplicial vertices, but in terms of other types of vertices that are used to characterize those classes.

Introducing subclasses of basic chordal graphs

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