The Clique Problem in Ray Intersection Graphs

  title={The Clique Problem in Ray Intersection Graphs},
  author={Sergio Cabello and Jean Cardinal and Stefan Langerman},
  journal={Discrete \& Computational Geometry},
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). 

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

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

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

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

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

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

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

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.

Maximum Clique in Disk-Like Intersection Graphs

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



Every planar graph is the intersection graph of segments in the plane: extended abstract

This paper proves a conjecture of Scheinerman that every planar graph is the intersection graph of some segments in the plane.

On intersection representations of co-planar graphs

Segment representation of a subclass of co-planar graphs

The Clique Problem in Intersection Graphs of Ellipses and Triangles

This work considers the Clique problem for intersection graphs of ellipses, which, in a sense, interpolate between disks and line segments, and shows that the problem is APX-hard in that case, and describes a simple approximation algorithm for the case of ellipsoidal graphs for which the ratio of radii is bounded.

The max clique problem in classes of string-graphs

The Rectilinear Steiner Tree Problem in NP Complete

The problem of determining the minimum length of an optimum rectilinear Steiner tree for a set A of points in the plane is shown to be NP-complete and the emphasis of the literature on heuristics and special case algorithms is well justified.

Independent set of intersection graphs of convex objects in 2D

Intersection Graphs of Segments

It is proved that the recognition of SEG-graphs is of the same complexity as the decision of solvability of a system of strict polynomial inequalities in the reals, i.e., as the decisions of a special existentially quantified sentence in the theory of real closed fields, and thus it belongs to PSPACE.

Computing the independence number of intersection graphs

A subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane, known to be an NP-hard task for general graphs.

Clique Is Hard to Approximate within n1-o(1)

The reductions used to prove Max Clique cannot be approximated in polynomial time within n1-Ɛ, for any constant Ɛ > 0, unless NP = ZPP are extended and combined with a recent result of Samorodnitsky and Trevisan.