On Contact Graphs of Paths on a Grid

Abstract

In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for B0-CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG.

Characterising circular-arc contact B0–VPG graphs

“…Since the hole starts and ends at the same vertex, and thus, with the same colour, the number of corners is even. As noticed in previous work [15,20], any contact B 0 -VPG representation of a K 4 necessarily contains a point where coincide one endpoint of each of the lines representing the four vertices. We say that this endpoint of the line is taken by the K 4 , which implies in particular that it cannot be the contact point with a line corresponding to a neighbour outside this K 4 .…”

“…In this paper, we focus solely on contact B 0 -VPG graphs. It was shown in [15,20] that recognising the class of contact B 0 -VPG graphs is NP-complete, and the complete list of minimal forbidden induced subgraphs for the class is not yet known. Nevertheless, characterisations of contact B 0 -VPG graphs by minimal forbidden induced subgraphs are known when restricted to some graph classes such as chordal, P 5 -free, P 4 -tidy, treecographs [6,7]; furthermore, most of those characterisations lead to polynomial-time recognition algorithms within the class.…”

CPG graphs: Some structural and hardness results

“…Although CPG graphs have been considered in the literature (see, e.g., [8,10,17,19,22,26]), a systematic study of this class was first initiated in [14]. We note that Aerts and Felsner [1] studied the family of graphs admitting a Vertex Contact representation of Paths on a Grid, which constitutes a subclass of CPG graphs as paths in such representations are not allowed to share a common endpoint.…”

“…We note that Aerts and Felsner [1] studied the family of graphs admitting a Vertex Contact representation of Paths on a Grid, which constitutes a subclass of CPG graphs as paths in such representations are not allowed to share a common endpoint. Such graphs are easily seen to be planar and in fact this class is strictly contained in that of planar CPG graphs [14]. It is then natural to ask whether every planar graph is CPG and whether there exists k ≥ 0 such that every planar CPG graph is B k -CPG.…”

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In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in [14]. A CPG graph is a graph for which there exists a collection of pairwise interiorly disjoint paths on a grid in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point.

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