Can visibility graphs be represented compactly?

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,..., 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 Σ_i^k=1 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 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 a size O(n log³ n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).