The Clique Problem in Ray Intersection Graphs

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 Kratochv́ıl and Nešetřil. ∗Research was supported in part by the Slovenian Research Agency, program P1-0297 and projects J1-4106, L74119, BI-BE/11-12-F-002, and within the EUROCORES Programme EUROGIGA (projects GReGAS and ComPoSe) of the European Science Foundation. Large part of the work was done while Sergio was visiting ULB. †Department of Mathematics, IMFM, and Department of Mathematics, FMF, University of Ljubljana, Slovenia. email: [email protected] ‡Université Libre de Bruxelles (ULB), Brussels, Belgium. Email: [email protected] §Mâıtre de Recherches, Fonds de la Recherche Scientifique (F.R.S-FNRS), Université Libre de Bruxelles (ULB), Brussels, Belgium. Email: [email protected] ar X iv :1 11 1. 59 86 v1 [ cs .C G ] 2 5 N ov 2 01 1