EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection
graphs of paths on an orthogonal grid. The class Bkβ-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
B1β-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 B2β and B3β-EPG graphs. In
this paper, we show that we can compute a maximum clique in polynomial time in
B2β-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 Bkβ-EPG graphs without
knowing the representation of the graph. It generalizes a result of [Epstein et
al, 2013] on B1β-EPG graphs (where the representation was needed)