 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2017.54
URN: urn:nbn:de:0030-drops-72101
URL: https://drops.dagstuhl.de/opus/volltexte/2017/7210/
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### From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

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### Abstract

A set P = H cup {w} of n+1 points 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 log 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^max(omega,k-2)) where omega = 2.373, even though the asymptotic number of facets may be as large as n^k.

### BibTeX - Entry

```@InProceedings{pilz_et_al:LIPIcs:2017:7210,
author =	{Alexander Pilz and Emo Welzl and Manuel Wettstein},
title =	{{From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices}},
booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
pages =	{54:1--54:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-038-5},
ISSN =	{1868-8969},
year =	{2017},
volume =	{77},
editor =	{Boris Aronov and Matthew J. Katz},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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