### 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 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{ω,k-2}) where ω ≈ 2.373, even though the asymptotic number of facets may be as large as n^{k}.

Original language | English |
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Title of host publication | 33rd International Symposium on Computational Geometry, SoCG 2017 |

Publisher | Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH |

Pages | 541-5416 |

Number of pages | 4876 |

Volume | 77 |

ISBN (Electronic) | 9783959770385 |

DOIs | |

Publication status | Published - 1 Jun 2017 |

Externally published | Yes |

Event | 33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia Duration: 4 Jul 2017 → 7 Jul 2017 |

### Conference

Conference | 33rd International Symposium on Computational Geometry, SoCG 2017 |
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Country | Australia |

City | Brisbane |

Period | 4/07/17 → 7/07/17 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review

*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

}

TY - GEN

T1 - From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices

AU - Pilz, Alexander

AU - Welzl, Emo

AU - Wettstein, Manuel

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

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.

KW - Gale transform

KW - Geometric graph

KW - Polytope

KW - Simplicial depth

KW - Wheel set

UR - http://www.scopus.com/inward/record.url?scp=85029928824&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2017.54

DO - 10.4230/LIPIcs.SoCG.2017.54

M3 - Conference contribution

VL - 77

SP - 541

EP - 5416

BT - 33rd International Symposium on Computational Geometry, SoCG 2017

PB - Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH

ER -