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

Valentin E. Brimkov; Konstanty Junosza-Szaniawski; Sean Kafer; Jan Kratochvíl; Martin Pergel; Paweł Rzążewski; Matthew Szczepankiewicz; Joshua Terhaar

arXiv · 1411.2928

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

Valentin E. Brimkov·Konstanty Junosza-Szaniawski·Sean Kafer·Jan Kratochvíl·Martin Pergel·Paweł Rzążewski·Matthew Szczepankiewicz·Joshua Terhaar

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Abstract

We study the {\sc 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, but solvable in polynomial time in intersection graphs of homothetic triangles. We extend the latter result by showing that for every convex polygon $P$ with sides parallel to $k$ directions, every $n$-vertex graph which is an intersection graph of homothetic copies of $P$ contains at most $n^{k}$ inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so called $k_{\text{DIR}}-\text{CONV}$, which are intersection graphs of convex polygons whose sides are parallel to some fixed $k$ directions. Moreover, we provide some lower bounds on the numbers of maximal cliques, discuss the complexity of recognizing these classes of graphs and present a relationship with other classes of convex-set intersection graphs. Finally, we generalize the upper bound on the number of maximal cliques to intersection graphs of higher-dimensional convex polytopes in Euclidean space.

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Valentin E. Brimkov, Konstanty Junosza-Szaniawski, Sean Kafer, Jan Kratochvíl, Martin Pergel, Paweł Rzążewski, Matthew Szczepankiewicz, Joshua Terhaar. 2014-11-11. Homothetic Polygons and Beyond: Intersection Graphs, Recognition, and Maximum Clique. https://arxiv.org/abs/1411.2928

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