Orientations of circle graphs

Let C(v1, …,vn) be a system consisting of a circle C with chords v1, …,vn on it having different endpoints. Define a graph G having vertex set V(G) = {v1, …,vn} and for which vertices vi and vj are adjacent in G if the chords vi and vj intersect. Such a graph will be called a circle graph. The chords divide the interior of C into a number of regions. We give a method which associates to each such region an orientation of the edges of G. For a given C(v1, …,vn) the number m of different orientations corresponding to it satisfies q + 1 ≤ m ≤ n + q + 1, where q is the number of edges in G. An oriented graph obtained from a diagram C(v1, …,vn) as above is called an oriented circle graph (OCG). We show that transitive orientations of permutation graphs are OCGs, and give a characterization of tournaments which are OCGs. When the region is a peripheral one, the orientation of G is acyclic. In this case we define a special orientation of the complement of G, and use this to develop an improved algorithm for finding a maximum independent set in G.