Can visibility graphs be represented compactly?

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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 Σki=1 ni, where ni is the number of vertices in Gi. Our main result is that there exist visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2n. An upper bound of 0(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(n log3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).