From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

A set P = H cup w of n+1 points in general position in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on such a set P, it suffices to know the frequency vector of P. While there are roughly 2n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2n/2. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, triangulations, and many more. Based on that, the corresponding numbers of graphs can be computed efficiently. In particular, we rediscover an already known formula for w-embracing triangles spanned by H. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of w-embracing simplices. While our previous arguments in the plane do not generalize easily, we show how to use similar ideas in ℝd for any fixed d. The result is an O(nd-1) time algorithm for computing the simplicial depth of a point w in a set H of n points, improving on the previously best bound of O(ndlog n). Based on our result about simplicial depth, we can compute the number of facets of the convex hull of n=d+k points in general position in ℝd in time O(n^maxω,k-2) where ω ≈ 2.373, even though the asymptotic number of facets may be as large as nk.