Discrete & Computational Geometry

, Volume 12, Issue 3, pp 347–365

Can visibility graphs Be represented compactly?

Authors

  • P. K. Agarwal
    • Department of Computer ScienceDuke University
  • N. Alon
    • Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv University
    • Bell Communications Research
  • B. Aronov
    • Computer Science DepartmentPolytechnic University, Six Metro Tech Center
  • S. Suri
    • Bell Communications Research
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