From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices
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A set P = H ∪{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 P, it suffices to know the so-called 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, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently. 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 simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. 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 d-dimensional points, improving on the previously best bound of O(n^d n). Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in R^d in time O(n^{ω,k-2}) where ω≈ 2.373, even though the asymptotic number of facets may be as large as n^k.
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