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 This research contains initial unpublished results about an exploration of the class B2- EPG, in addition to proposing interesting topics for future workA palavra EPG é um acrônimo para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

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

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k ≥ 0, the same way as Bk- EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circulararc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs

07211 Abstracts Collection -- Exact, Approximative, Robust and Certifying Algorithms on Particular Graph Classes

From May 20 to May 25, 2007, the Dagstuhl Seminar 07211 ``Exact, Approximative, Robust and Certifying Algorithms on Particular Graph Classes\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

K-SUN PERTENCE A B2-EPG-HELLY

In this article we explore the -EPG class and the Helly property. We present generic results on EPG representations and define terms that support the other results, in addition, we finish the research with an unpublished algorithm that builds a Helly -EPG representation of any k-sun graph.En este artículo exploramos la clase -EPG y la propiedad Helly. Presentamos resultados genéricos sobre las representaciones EPG y definimos términos que respaldan los otros resultados, además, finalizamos la investigación con un algoritmo original que construye una representación -EPG -Helly de cualquier grafo k-sun.Dans cet article, nous explorons la classe de graphes B_2-EPG et la propriété Helly. Nous présentons des résultats génériques sur les représentations EPG et définissons les termes qui supportent les autres résultats, en plus, nous présentons un algorithme sans précédent qui construit une représentation B_2-EPG-Helly de tout graphe k-sun.Neste artigo exploramos a classe de grafos -EPG e a propriedade Helly. Apresentamos resultados genéricos sobre representações EPG e definimos termos que suportam os demais resultados, além disso, apresentamos um algoritmo inédito que constrói uma representação -EPG-Helly de qualquer grafo k-sun

B2-EPG Split

On this research we study EPG graphs, in particular, we are interested on investigate the intersection between split and B2-EPGgraph classes. The results found in the literature regarding split graphs only concern about characterization in B1-EPG, that despite bringing some graphs that belong to this class, the bend number of split graphs it is still unknown. We study the split graphs whose degree of vertices on independent set is less than 2. In this work we manipulate bipartite graphs and we present representation for some bipartite graphs in B2-EPG. We build an algorithm that create a Split B2-EPG representation, also we present another algorithm that builds a representation on a grid Qw×2y+1 for any Split graph, and other results

Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

A graph $G$ is called an edge intersection graph of paths on a grid if there is a grid and there is a set of paths on this grid, such that the vertices of $G$ correspond to the paths and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge. Such a representation is called an EPG representation of $G$. $B_{k}$ is the class of graphs for which there exists an EPG representation where every path has at most $k$ bends. The bend number $b(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k$. $B_{k}^{m}$ is the subclass of $B_k$ containing all graphs for which there exists an EPG representation where every path has at most $k$ bends and is monotonic, i.e. it is ascending in both columns and rows. The monotonic bend number $b^m(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k^m$. Edge intersection graphs of paths on a grid were introduced by Golumbic, Lipshteyn and Stern in 2009 and a lot of research has been done on them since then. In this paper we deal with the monotonic bend number of outerplanar graphs. We show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize in terms of forbidden subgraphs the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$. As a consequence we show that for any maximal outerplanar graph and any cactus a (monotonic) EPG representation with the smallest possible number of bends can be constructed in a time which is polynomial in the number of vertices of the graph

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum