The clique problem in ray intersection graphs

Authors:
Sergio Cabello;Jean Cardinal;Stefan Langerman
Affiliations:
Department of Mathematics, IMFM, University of Ljubljana, Slovenia, Department of Mathematics, FMF, University of Ljubljana, Slovenia;Computer Science Department, Université Libre de Bruxelles (ULB), Brussels, Belgium;Computer Science Department, Université Libre de Bruxelles (ULB), Brussels, Belgium
Venue:
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Year:
2012

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Abstract

Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochvíl and Nešetřil.