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={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).