CPG graphs: Some structural and hardness results

Abstract In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in Deniz et al. (2018). 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. If every such path has at most k bends for some k ≥ 0 , the graph is said to be B k -CPG. We first show that, for any k ≥ 0 , the class of B k -CPG graphs is strictly contained in the class of B k + 1 -CPG graphs even within the class of planar graphs, thus implying that there exists no k ≥ 0 such that every planar CPG graph is B k -CPG. The main result of the paper is that recognizing CPG graphs and B k -CPG graphs with k ≥ 1 is NP -complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k ≥ 3 . We then consider several graph problems restricted to CPG graphs and show, in particular, that Independent Set and Clique Cover remain NP -hard for B 0 -CPG graphs. Finally, we consider the related classes B k -EPG of edge-intersection graphs of paths with at most k bends on a grid. Although it is possible to optimally color a B 0 -EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is NP -complete for B 1 -EPG graphs.