The icosahedron is clique divergent

AbstractA clique of a graph G is a maximal complete subgraph. The clique graph k(G) is the intersection graph of the set of all cliques of G. The iterated clique graphs are defined recursively by k0(G)=G and kn+1(G)=k(kn(G)). A graph G is said to be clique divergent (or k-divergent) if limn→∞|V(kn(G))|=∞. The problem of deciding whether the icosahedron is clique divergent or not was (implicitly) stated Neumann-Lara in 1981 and then cited by Neumann-Lara in 1991 and Larrión and Neumann-Lara in 2000. This paper proves the clique divergence of the icosahedron among other results of general interest in clique divergence theory

Clique graphs and Helly graphs

AbstractAmong the graphs for which the system of cliques has the Helly property those are characterized which are clique-convergent to the one-vertex graph. These graphs, also known as the so-called absolute retracts of reflexive graphs, are the line graphs of conformal Helly hypergraphs possessing a certain elimination scheme. From particular classes of such hypergraphs one can readily construct various classes G of graphs such that each member of G has its clique graph in G and is itself the clique graph of some other member of G. Examples include the classes of strongly chordal graphs and Ptolemaic graphs, respectively

Algebraic theory for the clique operator

In this text we attempt to unify many results about the K operator based on a new theory involving graphs, families and operators. We are able to build an "operator algebra" that helps to unify and automate arguments. In addition, we relate well-known properties, such as the Helly property, to the families and the operators. As a result, we deduce many classic results in clique graph theory from the basic fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Bandelt and Prisner's partial characterization of clique-fixed classes [2]. Furthermore, we show the power of our approach proving general results that lead to polynomial recognition of certain graph classes.Facultad de Ciencias Exacta

Modification Problems Toward Proper (Helly) Circular-Arc Graphs

We present a $9^k\cdot n^{O(1)}$-time algorithm for the proper circular-arc vertex deletion problem, resolving an open problem of van 't Hof and Villanger [Algorithmica 2013] and Crespelle et al. [arXiv:2001.06867]. Our structural study also implies parameterized algorithms for modification problems toward proper Helly circular-arc graphs

Subclasses of Normal Helly Circular-Arc Graphs

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled Proper Helly Circular-Arc Graphs) appeared at WG'0

On weighted clique graphs

Let K(G) be the clique graph of a graph G. A m-weighting of K(G) consists on giving to each m-size subset of its vertices a weight equal to the size of the intersection of the m corresponding cliques of G. The 2-weighted clique graph was previously considered by McKee. In this work we obtain a characterization of weighted clique graphs similar to Roberts and Spencer’s characterization for clique graphs. Some graph classes can be naturally defined in terms of their weighted clique graphs, for example clique-Helly graphs and their generalizations, and diamond-free graphs. The main contribution of this work is to characterize several graph classes by means of their weighted clique graph: hereditary clique-Helly graphs, split graphs, chordal graphs, UV graphs, interval graphs, proper interval graphs, trees, and block graphs.Sociedad Argentina de Informática e Investigación Operativ

Algebraic theory for the clique operator

In this text we attempt to unify many results about the K operator based on a new theory involving graphs, families and operators. We are able to build an "operator algebra" that helps to unify and automate arguments. In addition, we relate well-known properties, such as the Helly property, to the families and the operators. As a result, we deduce many classic results in clique graph theory from the basic fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Bandelt and Prisner's partial characterization of clique-fixed classes [2]. Furthermore, we show the power of our approach proving general results that lead to polynomial recognition of certain graph classes.Facultad de Ciencias Exacta

Grafos de intervalos propios y grafos arbóreos

Contenido: Introducción 1 Grafos de intervalos propios 1.1 Generalidades 1.2 Caracterizaciones 1.3 Radio y centro 1.4 Planaridad 1.5 Un problema de aplicación 2 Grafos de intervalos propios mínimos 2.1 Generalidades 2.2 Resultado Principal 2.3 Una clase clique-cerrada 2.4 Número de grafos de intervalos propios mínimos conexos 3 Grafos Arbóreo 3.1 Generalidades 3.2 Caracterizaciones 3.3 Relación con otras clases de grafos 4 Grafos de intersección 4.1 Generalidades 4.2 Una caracterización de los grafos de intersección 4.3 La aplicación dique entre ΩΣp y CΣpTesis digitalizada en SEDICI gracias a la Biblioteca del Departamento de Matemática de la Facultad de Ciencias Exactas (UNLP).Facultad de Ciencias Exacta

Grafos de intervalos propios y grafos arbóreos

Contenido: Introducción 1 Grafos de intervalos propios 1.1 Generalidades 1.2 Caracterizaciones 1.3 Radio y centro 1.4 Planaridad 1.5 Un problema de aplicación 2 Grafos de intervalos propios mínimos 2.1 Generalidades 2.2 Resultado Principal 2.3 Una clase clique-cerrada 2.4 Número de grafos de intervalos propios mínimos conexos 3 Grafos Arbóreo 3.1 Generalidades 3.2 Caracterizaciones 3.3 Relación con otras clases de grafos 4 Grafos de intersección 4.1 Generalidades 4.2 Una caracterización de los grafos de intersección 4.3 La aplicación dique entre ΩΣp y CΣpTesis digitalizada en SEDICI gracias a la Biblioteca del Departamento de Matemática de la Facultad de Ciencias Exactas (UNLP).Facultad de Ciencias Exacta