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 $k \geq 0$, 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. 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 $\mathsf{NP}$-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 $\mathsf{NP}$-complete for $B_1$-EPG graphs.

中文翻译：

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CPG 图：一些结构和硬度结果

在本文中，我们继续系统研究 [11] 中发起的网格上路径的接触图（CPG 图）。CPG 图是这样一种图，其中在网格上存在与其顶点集一一对应的成对内部不相交路径的集合，使得当且仅当相应的路径在网格点处接触时，两个顶点才相邻. 如果每条这样的路径至多有 $k$ 弯曲一些 $k \geq 0$，则该图被称为 $B_k$-CPG。我们表明，对于任何 $k \geq 0$，$B_k$-CPG 图的类严格包含在 $B_{k+1}$-CPG 图的类中，甚至在平面图的类中，因此暗示不存在 $k \geq 0$ 使得每个平面 CPG 图都是 $B_k$-CPG。此外，我们检查了几个仅限于 CPG 图的图问题的计算复杂性。特别是，我们表明，对于 $B_0$-CPG 图，独立集和 Clique Cover 仍然是 $\mathsf{NP}$-hard。最后，我们考虑在网格上最多具有 $k$ 弯曲的路径的边相交图的相关类 $B_k$-EPG。尽管可以在多项式时间内对 $B_0$-EPG 图进行最佳着色，但由于该类与区间图的类重合，我们表明，相比之下，3-Colorability 是 $\mathsf{NP}$-complete for $ B_1$-EPG 图表。