CPG graphs: Some structural and hardness results

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 B₀-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 B₀-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 B₁-EPG graphs.