In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in [ 12 ]. 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 \geq 0, the graph is said to be B_k-CPG. We show that for any k \geq 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 \geq 0 such that every planar CPG graph is B_k-CPG. We further show that Recognition is \mathsf{NP B_k-CPG  with k \geq -complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k \geq } 3. Additionally, we examine the computational complexity of several graph problems restricted to CPG graphs . In particular, we show 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.
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 \geq 0, the graph is said to be B_k-CPG. We first show that, for any k \geq 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 \geq 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 \geq 1 is \mathsf{NP-complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k \geq } 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.
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