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

Abstract

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 \(2^n\) distinct order types that correspond to wheel sets, the number of frequency vectors is only about \(2^{n/2}\). 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 \(\mathbb {R}^d\) for any fixed d. The result is an \(O(n^{d-1})\) 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 \(O(n^d\log n)\). 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 \(\mathbb {R}^d\) in time \(O(n^{\max \{\omega ,k-2\}})\) where \(\omega \approx 2.373\), even though the asymptotic number of facets may be as large as \(n^k\).