Beyond Homothetic Polygons: 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-sets intersection graphs and straight-line-segments 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 k sides, every n-vertex graph which is an intersection graph of homothetic copies of P contains at most n2k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, so called kDIR-CONV, which are intersection graphs of convex polygons whose all sides are parallel to at most k directions. We further provide lower bounds on the numbers of maximal cliques, discuss the complexity of recognizing these classes of graphs and present relationship with other classes of convex-sets intersection graphs.
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- 1.Badent, M., Binucci, C., Di Giacomo, E., Didimo, W., Felsner, S., Giordano, F., Kratochvíl, J., Palladino, P., Patrignani, M., Trotta, F.: Homothetic Triangle Contact Representations of Planar Graphs. In: Proc. CCCG 2007, pp. 233–236 (2007)Google Scholar
- 5.Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs. Acad. Press (1980)Google Scholar
- 8.Kaufmann, M., Kratochvíl, J., Lehmann, K., Subramanian, A.: Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition. In: Proc. SODA 2006, pp. 832–841 (2006)Google Scholar
- 16.Müller, T., van Leeuwen, E.J., van Leeuwen, J.: Integer representations of convex polygon intersection graphs. In: Symposium on Comput. Geometry, pp. 300–307 (2011)Google Scholar
- 17.Pergel, M.: Special graph classes and algorithms on them. Ph.D.-thesis, Charles University (2008)Google Scholar