On contact graphs of paths on a grid

TL;DR

This paper studies contact graphs of paths on a grid (CPG graphs), exploring their structural properties, colorability, recognition complexity, and their relation to planarity, revealing new bounds and distinctions within this graph class.

Contribution

It provides structural bounds, analyzes recognition and 3-colorability problems for CPG graphs, and clarifies their relationship with planar graphs, extending understanding of this graph class.

Findings

Constant upper bounds on clique and chromatic numbers for CPG graphs.

Recognition and 3-colorability problems are examined for a subclass of CPG.

CPG graphs are not necessarily planar, and not all planar graphs are CPG.

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 $B_0$-CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG.