Separation problems and circular arc systems

Abstract

We show that the problem of finding a smallest convex polygon which separates two given finite sets of points in the plane is a special case of the combinatorial problem of finding a minimum transversal of a circular arc system. We present an O(n log n) algorithm for the latter problem. We describe also a close relationship between visibility graphs and intersection graphs. It is furthermore shown that a smallest separating convex polygon is not greater than any separating arbitrary polygon or any separating planar subdivision. Moreover, we determine the number of stages needed for learning convex polygons from examples.