Can visibility graphs Be represented compactly?

P. K. Agarwal, N. Alon, B. Aronov, S. Suri

Research output: Contribution to journalArticle

44 Scopus citations

Abstract

We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graph G, a family G={G 1, G 2,..., G k } is called a clique cover of G if (i) each G i is a clique or a bipartite clique, and (ii) the union of G i is G. The size of the clique cover G is defined as ∑ i=1 k n i, where n i is the number of vertices in G i . Our main result is that there are visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size Ω(n 2/log2 n). An upper bound of O(n 2/log n) 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 size O(nlog3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).

Original languageEnglish (US)
Pages (from-to)347-365
Number of pages19
JournalDiscrete & Computational Geometry
Volume12
Issue number1
DOIs
StatePublished - Dec 1 1994
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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