Discrete & Computational Geometry

, Volume 12, Issue 3, pp 347–365

Can visibility graphs Be represented compactly?

  • P. K. Agarwal
  • N. Alon
  • B. Aronov
  • S. Suri
Article

DOI: 10.1007/BF02574385

Cite this article as:
Agarwal, P.K., Alon, N., Aronov, B. et al. Discrete Comput Geom (1994) 12: 347. doi:10.1007/BF02574385

Abstract

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={G1,G2,...,Gk} is called aclique cover ofG if (i) eachGi is a clique or a bipartite clique, and (ii) the union ofGi isG. The size of the clique coverG is defined as ∑i=1kni, whereni is the number of vertices inGi. Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2n). An upper bound ofO(n2/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(nlog3n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn).

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • P. K. Agarwal
    • 1
  • N. Alon
    • 2
    • 4
  • B. Aronov
    • 3
  • S. Suri
    • 4
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Computer Science DepartmentPolytechnic University, Six Metro Tech CenterBrooklynUSA
  4. 4.Bell Communications ResearchMorristownUSA