Planar segment visibility graphs

Given a set of n disjoint line segments in the plane, the segment visibility graph is the graph whose 2n vertices correspond to the endpoints of the line segments and whose edges connect every pair of vertices whose corresponding endpoints can see each other. In this paper we characterize and provide a polynomial time recognition algorithm for planar segment visibility graphs. Actually, we characterize segment visibility graphs that do not contain the complete graphK5 as a minor, and show that this class is the same as the class of planar segment visibility graphs. We use and prove the fact that every segment visibility graph containsK4 as a subgraph. In fact, we prove a stronger result: every set of n line segments determines at least n − 3 empty convex quadrilaterals.  2000 Elsevier Science B.V. All rights reserved.