# From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices

Alexander Pilz, Emo Welzl, Manuel Wettstein

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

### Abstract

A set P = H ⊃ {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 2n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2n/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(nd-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(nd 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 Rd in time O(nmax{ω,k-2) where ω ≈ 2.373, even though the asymptotic number of facets may be as large as nk.

Original language English 33rd International Symposium on Computational Geometry, SoCG 2017 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH 541-5416 4876 77 9783959770385 https://doi.org/10.4230/LIPIcs.SoCG.2017.54 Published - 1 Jun 2017 Yes 33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, AustraliaDuration: 4 Jul 2017 → 7 Jul 2017

### Conference

Conference 33rd International Symposium on Computational Geometry, SoCG 2017 Australia Brisbane 4/07/17 → 7/07/17

Wheels

### Keywords

• Gale transform
• Geometric graph
• Polytope
• Simplicial depth
• Wheel set

• Software

### Cite this

Pilz, A., Welzl, E., & Wettstein, M. (2017). From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices. In 33rd International Symposium on Computational Geometry, SoCG 2017 (Vol. 77, pp. 541-5416). Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. https://doi.org/10.4230/LIPIcs.SoCG.2017.54

From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices. / Pilz, Alexander; Welzl, Emo; Wettstein, Manuel.

33rd International Symposium on Computational Geometry, SoCG 2017. Vol. 77 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2017. p. 541-5416.

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Pilz, A, Welzl, E & Wettstein, M 2017, From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices. in 33rd International Symposium on Computational Geometry, SoCG 2017. vol. 77, Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, pp. 541-5416, 33rd International Symposium on Computational Geometry, SoCG 2017, Brisbane, Australia, 4/07/17. https://doi.org/10.4230/LIPIcs.SoCG.2017.54
Pilz A, Welzl E, Wettstein M. From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices. In 33rd International Symposium on Computational Geometry, SoCG 2017. Vol. 77. Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. 2017. p. 541-5416 https://doi.org/10.4230/LIPIcs.SoCG.2017.54
Pilz, Alexander ; Welzl, Emo ; Wettstein, Manuel. / From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices. 33rd International Symposium on Computational Geometry, SoCG 2017. Vol. 77 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2017. pp. 541-5416
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AB - A set P = H ⊃ {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 2n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2n/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(nd-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(nd 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 Rd in time O(nmax{ω,k-2) where ω ≈ 2.373, even though the asymptotic number of facets may be as large as nk.

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