Edge Intersection Graphs of L-Shaped Paths in Grids

In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: $\llcorner$,$\ulcorner$, $\urcorner$, $\lrcorner$, and we consider zero bend paths (i.e., | and $-$) to be degenerate $\llcorner$s. These graphs, called $B_1$-EPG graphs, were first introduced by Golumbic et al (2009). We consider the natural subclasses of $B_1$-EPG formed by the subsets of the four single bend shapes (i.e., {$\llcorner$}, {$\llcorner$,$\ulcorner$}, {$\llcorner$,$\urcorner$}, and {$\llcorner$,$\ulcorner$,$\urcorner$}) and we denote the classes by [$\llcorner$], [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$], and [$\llcorner$,$\ulcorner$,$\urcorner$] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [$\llcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$,$\urcorner$] $\subsetneq$ $B_1$-EPG; also, [$\llcorner$,$\ulcorner$] is incomparable with [$\llcorner$,$\urcorner$]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split $\cap$ [$\llcorner$].Comment: 14 pages, to appear in DAM special issue for LAGOS'1

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

Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} 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 $k$, $B_k$-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 $B_4$-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is $B_3$-EPG, and that there exist circular-arc graphs which are not $B_2$-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define $B_k$-EPR graphs, $k\geq 0$, the same way as $B_k$-EPG graphs. Circular-arc graphs are clearly $B_4$-EPR graphs and we will show that there exist circular-arc graphs that are not $B_3$-EPR graphs. We also show that normal circular-arc graphs are $B_2$-EPR graphs and that there exist normal circular-arc graphs that are not $B_1$-EPR graphs. Finally, we characterize $B_1$-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs

Intersection Graphs of L-Shapes and Segments in the Plane

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: ⌊,⌈,⌋ and ⌉. A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an ⌊, an ⌊ or ⌈, a k-bend path, or a segment, then this graph is called an ⌊-graph, ⌊,⌈-graph, B k -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365–372, 1992], stating that every ⌊,⌈-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are ⌊-graphs, or B k -VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are ⌊-graphs. Furthermore we show that all complements of planar graphs are B 19-VPG-graphs and all complements of full subdivisions are B 2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once

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