Ray shooting from convex ranges

Consider a set X of n-points in space, each with positive z-coordinate, and a compact plane convex set S. We define a graph G(X,S), called the ray-sho oting graph, on X as follows: The points of X are the vertices and {p(i), p (j)}, p(i), p(j) is an element of X, is an edge if and only if the infinite line joining p(i) to p(j) intersects S. In this paper we provide a charact erization of shooting graphs; they are exactly the set of comparability gra phs of containment orders arising from families of plane homothetic convex sets. We also prove that the crossing number of these ordered sets is at mo st 2, and that not all comparability graphs are three-dimensional ray-shoot ing graphs. (C) 2001 Elsevier Science B.V. All rights reserved. MSC. 68R10; 68U05.