Crossing-free configurations on planar point sets

Abstract

This thesis treats crossing-free geometric graphs, which are graphs defined on a given finite set of points in the plane. Their edges are understood as straight-line segments connecting corresponding endpoints without passing through any other point. For a crossing-free graph no two edges are allowed to intersect other than in a common endpoint. The following are the most important kinds of crossing-free configurations that we will encounter in our discussion: Triangulations, crossing-free partitions, spanning trees, perfect matchings, and the set of all plane graphs. First, we consider a class of plane graphs that emerge from crossing-free partitions of the underlying point set. To be more precise, a partition of a given set of points is crossing-free if the convex hulls of the individual parts are mutually disjoint. Each partition naturally translates to a plane graph whose vertices are the given points and whose edges are the boundaries of the convex hulls of the partition classes. We ask whether convex position of the underlying point set minimizes the number of crossing-free partitions over all placements of equally many points. We answer a corresponding question in the affirmative for the number of crossing-free partitions of n points into a fixed number k of parts, where k ∈ {1, 2, 3, n−3, n−2, n−1, n}. In addition, we show that on at least five points the number of crossingfree partitions is not maximized in convex position. It is known that in convex position the number of crossing-free partitions into k classes equals the number of partitions into n − k + 1 parts. This does not hold in general, and we mention a construction for point sets with significantly more partitions into few classes than into many. Another problem we consider on point sets in convex position is the decomposition of the complete graph using geometric graphs corresponding to crossing-free partitions. We show almost tight bounds for the number of elements in a smallest possible decomposition. Second, we treat transformation graphs of crossing-free configurations on a set of points. These are abstract graphs whose vertices are the crossingfree configurations of interest and whose edges are defined by a prescribed