Computing Maximum Cliques in B_2 -EPG Graphs

EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection graphs of paths on an orthogonal grid. The class Bk-EPG is the subclass of EPG graphs where the path on the grid associated to each vertex has at most k bends. Epstein et al. showed in 2013 that computing a maximum clique in B1-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the number of bends is at least 4, the class contains 2-interval graphs for which computing a maximum clique is an NP-hard problem. The complexity status of the Maximum Clique problem remains open for B2 and B3-EPG graphs. In this paper, we show that we can compute a maximum clique in polynomial time in B2-EPG graphs given a representation of the graph. Moreover, we show that a simple counting argument provides a 2(k + 1)-approximation for the coloring problem on Bk-EPG graphs without knowing the representation of the graph. It generalizes a result of [Epstein et al, 2013] on B1-EPG graphs (where the representation was needed).