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 and Computational Geometry
    Volume12
    Issue number1
    DOIs
    StatePublished - Dec 1994

    Fingerprint

    Visibility Graph
    Clique
    Visibility
    Cover
    Graph theory
    Simple Polygon
    Line segment
    Union
    Extremal Graph Theory
    Upper bound

    ASJC Scopus subject areas

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

    Cite this

    Can visibility graphs Be represented compactly? / Agarwal, P. K.; Alon, N.; Aronov, B.; Suri, S.

    In: Discrete and Computational Geometry, Vol. 12, No. 1, 12.1994, p. 347-365.

    Research output: Contribution to journalArticle

    Agarwal, PK, Alon, N, Aronov, B & Suri, S 1994, 'Can visibility graphs Be represented compactly?', Discrete and Computational Geometry, vol. 12, no. 1, pp. 347-365. https://doi.org/10.1007/BF02574385
    Agarwal, P. K. ; Alon, N. ; Aronov, B. ; Suri, S. / Can visibility graphs Be represented compactly?. In: Discrete and Computational Geometry. 1994 ; Vol. 12, No. 1. pp. 347-365.
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