### Abstract

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Ω(2.631^{n}) geometric triangulations. Our result improves the previously best general lower bound of Ω(2.43^{n}) and also covers the previously best lower bound of Ω(2.63^{n}) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: 1. Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. 2. Generalized configurations of points that minimize the number of triangulations have at most ⌊n/2⌋ points on their convex hull. 3. We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.

Original language | English |
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Title of host publication | 32nd International Symposium on Computational Geometry, SoCG 2016 |

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

Pages | 7.1-7.16 |

Volume | 51 |

ISBN (Electronic) | 9783959770095 |

DOIs | |

Publication status | Published - 1 Jun 2016 |

Event | 32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States Duration: 14 Jun 2016 → 17 Jun 2016 |

### Conference

Conference | 32nd International Symposium on Computational Geometry, SoCG 2016 |
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Country | United States |

City | Boston |

Period | 14/06/16 → 17/06/16 |

### Fingerprint

### Keywords

- Combinatorial geometry
- Order types
- Triangulations

### ASJC Scopus subject areas

- Software

### Cite this

*32nd International Symposium on Computational Geometry, SoCG 2016*(Vol. 51, pp. 7.1-7.16). Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. https://doi.org/10.4230/LIPIcs.SoCG.2016.7

**An improved lower bound on the minimum number of triangulations.** / Aichholzer, Oswin; Alvarez, Victor; Hackl, Thomas; Pilz, Alexander; Speckmann, Bettina; Vogtenhuber, Birgit.

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

*32nd International Symposium on Computational Geometry, SoCG 2016.*vol. 51, Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, pp. 7.1-7.16, 32nd International Symposium on Computational Geometry, SoCG 2016, Boston, United States, 14/06/16. https://doi.org/10.4230/LIPIcs.SoCG.2016.7

}

TY - GEN

T1 - An improved lower bound on the minimum number of triangulations

AU - Aichholzer, Oswin

AU - Alvarez, Victor

AU - Hackl, Thomas

AU - Pilz, Alexander

AU - Speckmann, Bettina

AU - Vogtenhuber, Birgit

PY - 2016/6/1

Y1 - 2016/6/1

N2 - Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Ω(2.631n) geometric triangulations. Our result improves the previously best general lower bound of Ω(2.43n) and also covers the previously best lower bound of Ω(2.63n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: 1. Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. 2. Generalized configurations of points that minimize the number of triangulations have at most ⌊n/2⌋ points on their convex hull. 3. We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.

AB - Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Ω(2.631n) geometric triangulations. Our result improves the previously best general lower bound of Ω(2.43n) and also covers the previously best lower bound of Ω(2.63n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: 1. Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. 2. Generalized configurations of points that minimize the number of triangulations have at most ⌊n/2⌋ points on their convex hull. 3. We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.

KW - Combinatorial geometry

KW - Order types

KW - Triangulations

UR - http://www.scopus.com/inward/record.url?scp=84976902795&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2016.7

DO - 10.4230/LIPIcs.SoCG.2016.7

M3 - Conference contribution

VL - 51

SP - 7.1-7.16

BT - 32nd International Symposium on Computational Geometry, SoCG 2016

PB - Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH

ER -