The Clique Problem in Ray Intersection Graphs
Sergio Cabello, Jean Cardinal, Stefan Langerman
TL;DR
This paper proves that finding maximum cliques in segment intersection graphs is NP-hard by showing that any planar graph can be transformed into a ray intersection graph, solving a long-standing open problem.
Contribution
It demonstrates that the maximum clique problem in segment intersection graphs is NP-hard, resolving a 21-year-old open problem.
Findings
Any planar graph has an even subdivision whose complement is a ray intersection graph.
The construction is polynomial-time.
The result implies NP-hardness of maximum clique in segment intersection graphs.
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 Kratochv\'il and Ne\v{s}et\v{r}il.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · VLSI and FPGA Design Techniques · Advanced Graph Theory Research