Beyond Homothetic Polygons: Recognition and Maximum Clique

  • Konstanty Junosza-Szaniawski
  • Jan Kratochvíl
  • Martin Pergel
  • Paweł Rzążewski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)


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 n 2k inclusion-wise maximal cliques. We actually prove this result for a more general class of graphs, so called k DIR -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.


Convex Polygon Maximal Clique Intersection Graph Interval Graph Sweeping Line 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Konstanty Junosza-Szaniawski
    • 1
  • Jan Kratochvíl
    • 2
  • Martin Pergel
    • 3
  • Paweł Rzążewski
    • 1
  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarszawaPoland
  2. 2.Department of Applied Mathematics, and Institute for Theoretical Computer ScienceCharles UniversityPraha 1Czech Republic
  3. 3.Department of Software and Computer Science EducationCharles UniversityPraha 1Czech Republic

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