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 $2^n$ 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 $n^k$.