Hardness and approximation for L-EPG and B₁-EPG graphs

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.