Can visibility graphs Be represented compactly?

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={G1, G2,..., Gk } is called a clique cover of G if (i) each Gi is a clique or a bipartite clique, and (ii) the union of Gi is G. The size of the clique cover G is defined as ∑i=1k ni, where ni is the number of vertices in Gi . Our main result is that there are visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2 n). An upper bound of O(n2/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). © 1994 Springer-Verlag New York Inc.