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 Bk-CPG.We first show that, for any k≥0, the class of Bk-CPG graphs is strictly contained in the class of Bk+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 Bk-CPG. The main result of the paper is that recognizing CPG graphs and Bk-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 Bk-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.
Highlights
Golumbic et al [21] introduced the class of Edge intersection graphs of Paths on a Grid (EPG for short) that is, graphs for which there exists a collection of 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 intersect on at least one grid-edge
The end-eating graph E1 depicted in Fig. 6 will be used in the proof of the NP-hardness of Recognition restricted to B1-CPG graphs; the end-eating graph E2 depicted in Fig. 7 will be used in the proof of the NP-hardness of Recognition restricted to B2-CPG graphs; and the end-eating graph E3 depicted in Fig. 8 will be used in the proof of the NP-hardness of Recognition restricted to planar Bk-CPG graphs, for k ≥ 3
We showed that for any k ≥ 0, the class of Bk-CPG graphs is strictly contained in that of Bk+1-CPG, even within the class of planar graphs
Summary
Golumbic et al [21] introduced the class of Edge intersection graphs of Paths on a Grid (EPG for short) that is, graphs for which there exists a collection of 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 intersect on at least one grid-edge. Asinowski et al [4] provided several complexity results for problems restricted to B0-VPG graphs They showed that Independent Set, Hamiltonian Cycle, Hamiltonian Path and Colorability are all NP-complete, whereas Clique is polynomial-time solvable. The fact that every planar bipartite graph is B0-CPG [14] immediately implies that problems which are NP-complete for this class, such as Dominating Set, Feedback Vertex Set, Hamiltonian Cycle and Hamiltonian Path (see, e.g., [11,31]), remain NP-complete in B0-CPG. It was shown in [15] that 3-Colorability is NP-complete in B0-CPG.
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