Computing a maximum clique in geometric superclasses of disk graphs

TL;DR

This paper introduces a polynomial-time algorithm for a geometric superclass of unit disk graphs to compute maximum cliques and provides partial results towards an EPTAS for intersection graphs of convex pseudo-disks, advancing understanding in geometric intersection graph algorithms.

Contribution

The paper presents a polynomial-time algorithm for a superclass of unit disk graphs and partial progress towards an EPTAS for convex pseudo-disk intersection graphs.

Findings

Polynomial-time algorithm for a geometric superclass of unit disk graphs.

Partial results towards an EPTAS for convex pseudo-disk intersection graphs.

Clarification of complexity boundaries in geometric intersection graph problems.

Abstract

In the 90's Clark, Colbourn and Johnson wrote a seminal paper where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of d-dimensional (unit) balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs. However, the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.