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 $B_k$-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 $B_1$-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 $B_2$ and $B_3$-EPG graphs. In this paper, we show that we can compute a maximum clique in polynomial time in $B_2$-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 $B_k$-EPG graphs without knowing the representation of the graph. It generalizes a result of [Epstein et al, 2013] on $B_1$-EPG graphs (where the representation was needed).