Hardness and approximation for L-EPG and [formula omitted]-EPG graphs

Abstract

The edge intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. We consider the case of single-bend paths and its subclass of ⌞-shaped paths, namely, the classes known as B1-EPG and ⌞-EPG graphs, respectively.We show that fully-subdivided graphs are ⌞-EPG graphs, and later use this result in order to show that the following problems are APX-hard on ⌞-EPG graphs: Minimum Vertex Cover, Maximum Independent Set, and Maximum Weighted Independent Set, and also that Minimum Dominating Set is NP-complete on ⌞-EPG graphs. We also observe that Minimum Coloring is NP-complete already on ⌞-EPG, which follows from a proof for B1-EPG in Epstein et al. (2013). Finally, we provide efficient constant-factor-approximation algorithms for each of these problems on B1-EPG graphs.