Computing maximum cliques in B 2 EPG graphs

Nicolas Bousquet (1) , Marc Heinrich (2)

1 G-SCOP_OC - Optimisation Combinatoire
2 GOAL - Graphes, AlgOrithmes et AppLications

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 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 B k-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).

Mots clés: EPG Graphs; Maximum clique problem; Graph Theory

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Nicolas Bousquet, Marc Heinrich. Computing maximum cliques in B 2 EPG graphs. WG: Workshop on Graph-Theoretic Concepts in Computer Science, Jun 2017, Eindhoven, Netherlands. ⟨hal-01557335⟩