Computing Maximum Cliques in $$B_2$$ -EPG Graphs

Abstract

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 , 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).

On $k$-Bend and Monotonic $\ell$-Bend Edge Intersection Graphs of Paths on a Grid

“…Subject of investigation are certain NP-hard combinatorial optimization problems which turn out to be tractable, i.e. polynomially solvable or approximable within a guaranteed approximation ratio, for B k -EPG graphs, see [5][6][7]12]. Thus the computation of the bend number and the monotonic bend number of graphs or related upper bounds is a relevant research question in this context.…”