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 \(B_1\)-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).
N. Bousquet—Supported by ANR Projects STINT (anr-13-bs02-0007) and LabEx PERSYVAL-Lab (anr-11-labx-0025-01).
M. Heinrich—Supported by the anr-14-ce25-0006 project of the French National Research Agency.
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Bousquet, N., Heinrich, M. (2017). Computing Maximum Cliques in \(B_2\)-EPG Graphs. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_11
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DOI: https://doi.org/10.1007/978-3-319-68705-6_11
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