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 crossingfree geometric graphs on P, it suffices to know the socalled 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 crossingfree spanning cycles, matchings, wembracing 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^{d1}) time algorithm for computing the simplicial depth of a point w in a set H of n ddimensional 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 highdimensional 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,k2)) 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 CrossingFree Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices}},
booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)},
pages = {54:154:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770385},
ISSN = {18688969},
year = {2017},
volume = {77},
editor = {Boris Aronov and Matthew J. Katz},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7210},
URN = {urn:nbn:de:0030drops72101},
doi = {10.4230/LIPIcs.SoCG.2017.54},
annote = {Keywords: Geometric Graph, Wheel Set, Simplicial Depth, Gale Transform, Polytope}
}
Keywords: 

Geometric Graph, Wheel Set, Simplicial Depth, Gale Transform, Polytope 
Collection: 

33rd International Symposium on Computational Geometry (SoCG 2017) 
Issue Date: 

2017 
Date of publication: 

20.06.2017 