Discrete & Computational Geometry

, Volume 12, Issue 3, pp 347-365

First online:

Can visibility graphs Be represented compactly?

  • P. K. AgarwalAffiliated withDepartment of Computer Science, Duke University
  • , N. AlonAffiliated withDepartment of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv UniversityBell Communications Research
  • , B. AronovAffiliated withComputer Science Department, Polytechnic University, Six Metro Tech Center
  • , S. SuriAffiliated withBell Communications Research

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We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graphG, a familyG={G 1,G 2,...,G k } is called aclique cover ofG if (i) eachG i is a clique or a bipartite clique, and (ii) the union ofG i isG. The size of the clique coverG is defined as ∑ i=1 k n i , wheren i is the number of vertices inG i . Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n 2/log2 n). An upper bound ofO(n 2/logn) on the clique cover follows from a well-known result in extremal graph theory. On the other hand, we show that the visibility graph of a simple polygon always admits a clique cover of sizeO(nlog3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn).