On the helly property of some intersection graphs

An EPG graph G is an edge-intersection graph of paths on a grid. In this doctoral thesis we will mainly explore the EPG graphs, in particular B1-EPG graphs. However, other classes of intersection graphs will be studied such as VPG, EPT and VPT graph classes, in addition to the parameters Helly number and strong Helly number to EPG and VPG graphs. We will present the proof of NP-completeness to Helly-B1-EPG graph recognition problem. We investigate the parameters Helly number and the strong Helly number in both graph classes, EPG and VPG in order to determine lower bounds and upper bounds for this parameters. We completely solve the problem of determining the Helly and strong Helly numbers, for Bk-EPG, and Bk-VPG graphs, for each value k. Next, we present the result that every Chordal B1-EPG graph is simultaneously in the VPT and EPT graph classes. In particular, we describe structures that occur in B1-EPG graphs that do not support a Helly-B1-EPG representation and thus we define some sets of subgraphs that delimit Helly subfamilies. In addition, features of some non-trivial graph families that are properly contained in Helly-B1 EPG are also presented.EPG é um grafo de aresta-interseção de caminhos sobre uma grade. Nesta tese de doutorado exploraremos principalmente os grafos EPG, em particular os grafos B1-EPG. Entretanto, outras classes de grafos de interseção serão estu dadas, como as classes de grafos VPG, EPT e VPT, além dos parâmetros número de Helly e número de Helly forte nos grafos EPG e VPG. Apresentaremos uma prova de NP-completude para o problema de reconhecimento de grafos B1-EPG Helly. Investigamos os parâmetros número de Helly e o número de Helly forte nessas duas classes de grafos, EPG e VPG, a fim de determinar limites inferiores e superi ores para esses parâmetros. Resolvemos completamente o problema de determinar o número de Helly e o número de Helly forte para os grafos Bk-EPG e Bk-VPG, para cada valor k. Em seguida, apresentamos o resultado de que todo grafo B1-EPG Chordal está simultaneamente nas classes de grafos VPT e EPT. Em particular, descrevemos estruturas que ocorrem em grafos B1-EPG que não suportam uma representação B1-EPG-Helly e assim definimos alguns conjuntos de subgrafos que delimitam sub famílias Helly. Além disso, também são apresentadas características de algumas famílias de grafos não triviais que estão propriamente contidas em B1-EPG-Hell

Um estudo sobre grafos B2-EPG e B2-EPG-Helly

The word EPG is an acronym for Edge-Intersecting Paths on a Grid, that is, it exactly represents the class of edge-intersecion graphs of paths on a grid. In this writing, we started exploring the EPG graph subclass, well known as B2-EPG-Helly (more specifically its recognition complexity). However, we have also investigated graph representations that are not B1-EPG, but have not yet been associated with any Bk-EPG class, and also study other path properties in B2-EPG and B2-EPG with the Helly property. This research contains initial unpublished results about an exploration of the class B2- EPG, in addition to proposing interesting topics for future workA palavra EPG é um acrônimo para Edge-intersection Paths on a Grid, isto é, representa exatamente a classe de grafos de aresta-interseção de caminhos sobre uma grade. Neste trabalho de conclusão de curso iremos explorar principalmente uma subclasse de grafos EPG, conhecida como B2-EPG-Helly (mais especificamente a sua complexidade de reconhecimento). Contudo, também investigamos representações de grafos que não são B1-EPG, mas ainda não foram associadas a alguma classe Bk- EPG, além de estudar outras propriedades de caminhos em B2-EPG e B2-EPG com a propriedade Helly. Essa pesquisa contém resultados iniciais inéditos sobre a exploração da classe B2- EPG, além de propor tópicos interessantes para trabalhos futuro

Essential obstacles to Helly circular-arc graphs

A Helly circular-arc graph is the intersection graph of a set of arcs on a circle having the Helly property. We introduce essential obstacles, which are a refinement of the notion of obstacles, and prove that essential obstacles are precisely the minimal forbidden induced circular-arc subgraphs for the class of Helly circular-arc graphs. We show that it is possible to find in linear time, in any given obstacle, some minimal forbidden induced subgraph for the class of Helly circular-arc graphs contained as an induced subgraph. Moreover, relying on an existing linear-time algorithm for finding induced obstacles in circular-arc graphs, we conclude that it is possible to find in linear time an induced essential obstacle in any circular-arc graph that is not a Helly circular-arc graph. The problem of finding a forbidden induced subgraph characterization, not restricted only to circular-arc graphs, for the class of Helly circular-arc graphs remains unresolved. As a partial answer to this problem, we find the minimal forbidden induced subgraph characterization for the class of Helly circular-arc graphs restricted to graphs containing no induced claw and no induced 5-wheel. Furthermore, we show that there is a linear-time algorithm for finding, in any given graph that is not a Helly circular-arc graph, an induced subgraph isomorphic to claw, 5-wheel, or some minimal forbidden induced subgraph for the class of Helly circular-arc graphs.Fil: Safe, Martin Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin