# 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).

## 38 Citations

### Homothetic Polygons and Beyond: Intersection Graphs, Recognition, and Maximum Clique

- MathematicsArXiv
- 2014

We study the Clique problem in classes of intersection graphs of convex sets in the plane. The problem is known to be NP-complete in convex-set intersection graphs and straight-line-segment…

### Intersection Graphs of Rays and Grounded Segments

- MathematicsWG
- 2017

Several classes of intersection graphs of line segments in the plane are considered and it is shown that not every intersection graph of rays is an intersections graph of downward rays, and not every outer segment graph is an intersectiongraph of rays.

### Beyond Homothetic Polygons: Recognition and Maximum Clique

- MathematicsISAAC
- 2012

It is shown that for every convex polygon P with k sides, every n-vertex graph which is an intersection graph of homothetic copies of P contains at most n 2k inclusion-wise maximal cliques.

### On Intersection Graphs of Convex Polygons

- Mathematics, Computer ScienceIWCIA
- 2014

It is proved that the number of maximal cliques does not exceed n k, and it is shown that this bound is tight for any fixed k.

### Partitioning Geometric Graphs into Plane Subgraphs

- 2022

A geometric graph is a set of points in the plane with connecting straight-line segments. This thesis examines the problem of partitioning geometric graphs into subgraphs without intersecting…

### Knapsack Intersection Graphs and Efficient Computation of Their Maximal Cliques

- Mathematics, Computer ScienceCompIMAGE
- 2014

If the linear constraints defining the knapsack polygons are known, then the maximal clique problem onknapsack graphs can be solved using the algorithm from [28], and it is shown how these can be found efficiently in computation time bounded by a low degree polynomial in the polygons size.

### Planar point sets determine many pairwise crossing segments

- MathematicsSTOC
- 2019

It is shown that any set of n points in general position in the plane determines n1−o(1) pairwise crossing segments and the proof is fully constructive, and extends to dense geometric graphs.

### Homothetic polygons and beyond: Maximal cliques in intersection graphs

- MathematicsDiscret. Appl. Math.
- 2018

### Maximum Clique in Disk-Like Intersection Graphs

- MathematicsFSTTCS
- 2020

A scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP- hardness for (convex) pseudo-disks.

### Intersection graphs of L-shapes and segments in the plane

- Mathematics, GeologyDiscret. Appl. Math.
- 2014

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