93-59 Can Visibility Graphs Be Represented Compactly?

DIMACS TR: 93-59

Can Visibility Graphs Be Represented Compactly?

Authors: Pankaj K. Agarwal, Noga Alon, Boris Aronov, Subhash Suri

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 ${\cal G} = \{G_1, G_2, \ldots, G_k\}$ 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 {\cal G} is defined as \sum_{i=1}^k n_i, where n_i is the number of vertices in G_i. Our main result is that there exist visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size \Omega(n^2/\log^2 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(n \log^3 n), and that there are simple polygons whose visibility graphs require a clique cover of size \Omega (n \log n).

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-59.ps

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