CPG graphs: Some structural and hardness results
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In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in [11]. 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 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. 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.
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