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, 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 nvertex graph which is an intersection graph of homothetic copies of P contains at most n inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, the so called kDIR-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 maximum number of maximal cliques, discuss the complexity of recognizing these classes of graphs and present relationships 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. ∗Mathematics Department, SUNY Buffalo State, Buffalo, NY 14222, USA. †Warsaw University of Technology, Faculty of Mathematics and Information Science, Koszykowa 75, 00-662 Warszawa, Poland. ‡Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. §Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic. ¶Department of Software and Computer Science Education, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic. ‖Mathematics Department, University at Buffalo, Buffalo, NY 14260-2900, USA. 1 ar X iv :1 41 1. 29 28 v2 [ cs .D M ] 1 4 N ov 2 01 6