Clique Cover on L-EPG representations of graphs

Abstract

Clique Cover is a classical graph theory problem where we are given a graph $G=(V,E)$, an integer $k$, and asked if $V\left(G\right)$ can be partitioned into at most $k$ cliques. Edge intersection graphs of paths in grids (EPG graphs) are graphs whose vertices can be represented as nontrivial paths in a grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. When the paths have at most one change of direction (bend), these graphs are called $B_1$-EPG graphs; in addition, they are called $⌞$-EPG graphs (or $⌞$-shaped graphs) if all the paths are represented with one of the following shapes: “$⌞$”, “–”, or “$\mid$”. Such a representation is called $⌞$-EPG representation. The class of $⌞$-EPG graphs is a natural superclass of interval graphs. Since all maximal cliques of a $B_1$-EPG graph $G$ can be computed in polynomial time, and Clique Cover on interval graphs is polynomial-time solvable; in this paper, we study the complexity of Clique Cover on $⌞$-EPG graphs. We show that given an $⌞$-EPG representation of a graph $G$, it is NP-complete to determine whether $V\left(G\right)$ can be partitioned into at most $k$ cliques, but it can be solved in FPT time when parameterized by $k$. Furthermore, we adapt our algorithm to the case where the input is a graph without its $⌞$-EPG representation. Finally, we also show that Clique Cover on $\partial$EPG${}_2$ representations (a particular class of $⌞$-EPG representations) is polynomial-time solvable.