The Clique Problem in Ray Intersection Graphs

@article{Cabello2013TheCP,
  title={The Clique Problem in Ray Intersection Graphs},
  author={Sergio Cabello and Jean Cardinal and Stefan Langerman},
  journal={Discrete \& Computational Geometry},
  year={2013},
  volume={50},
  pages={771-783}
}
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 (Comment Math Univ Carolinae 31(1):85–93, 1990). 

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