# CPG graphs: Some structural and hardness results

## TL;DR

This paper investigates the structural properties and computational complexity of Contact graphs of Paths on a Grid (CPG graphs), establishing their hierarchy, NP-completeness of recognition problems, and complexity of classic graph problems within these classes.

## Contribution

It proves strict containment among classes of CPG graphs with bounded bends, and shows recognition and certain problem complexities are NP-complete even for planar graphs.

## Key findings

- $B_k$-CPG classes are strictly contained in $B_{k+1}$-CPG classes.
- Recognition of CPG and $B_k$-CPG graphs is NP-complete for $k \\geq 1$.
- NP-hardness of Independent Set and Clique Cover on $B_0$-CPG graphs.

## 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 $k \geq 0$, 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 $\mathsf{NP}$-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.