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

TL;DR

This paper investigates the complexity and properties of intersection graphs of convex polygons, extending known polynomial-time results for homothetic triangles to broader classes and higher dimensions, and analyzing the number of maximal cliques.

Contribution

It generalizes the polynomial-time recognition results from homothetic triangles to convex polygons with sides in fixed directions and higher-dimensional polytopes, providing bounds on maximal cliques.

Findings

Maximum number of maximal cliques is bounded by n^k for polygons with sides in k directions.

Recognition complexity varies across classes of convex-set intersection graphs.

Upper bounds on maximal cliques extend to higher-dimensional convex polytopes.

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.