Can visibility graphs Be represented compactly?

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

    Research output: Contribution to journalArticle

    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

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    ASJC Scopus subject areas

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

    Cite this

    Agarwal, P. K., Alon, N., Aronov, B., & Suri, S. (1994). Can visibility graphs Be represented compactly? Discrete & Computational Geometry, 12(1), 347-365. https://doi.org/10.1007/BF02574385