An improved lower bound on the minimum number of triangulations

Oswin Aichholzer, Victor Alvarez, Thomas Hackl, Alexander Pilz, Bettina Speckmann, Birgit Vogtenhuber

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandForschungBegutachtung

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(2.631n) 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.

Originalspracheenglisch
Titel32nd International Symposium on Computational Geometry, SoCG 2016
Herausgeber (Verlag)Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH
Seiten7.1-7.16
Band51
ISBN (elektronisch)9783959770095
DOIs
PublikationsstatusVeröffentlicht - 1 Jun 2016
Veranstaltung32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, USA / Vereinigte Staaten
Dauer: 14 Jun 201617 Jun 2016

Konferenz

Konferenz32nd International Symposium on Computational Geometry, SoCG 2016
LandUSA / Vereinigte Staaten
OrtBoston
Zeitraum14/06/1617/06/16

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    Aichholzer, O., Alvarez, V., Hackl, T., Pilz, A., Speckmann, B., & Vogtenhuber, B. (2016). An improved lower bound on the minimum number of triangulations. in 32nd International Symposium on Computational Geometry, SoCG 2016 (Band 51, S. 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.

    32nd International Symposium on Computational Geometry, SoCG 2016. Band 51 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2016. S. 7.1-7.16.

    Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem KonferenzbandForschungBegutachtung

    Aichholzer, O, Alvarez, V, Hackl, T, Pilz, A, Speckmann, B & Vogtenhuber, B 2016, An improved lower bound on the minimum number of triangulations. in 32nd International Symposium on Computational Geometry, SoCG 2016. Bd. 51, Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, S. 7.1-7.16, Boston, USA / Vereinigte Staaten, 14/06/16. https://doi.org/10.4230/LIPIcs.SoCG.2016.7
    Aichholzer O, Alvarez V, Hackl T, Pilz A, Speckmann B, Vogtenhuber B. An improved lower bound on the minimum number of triangulations. in 32nd International Symposium on Computational Geometry, SoCG 2016. Band 51. Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH. 2016. S. 7.1-7.16 https://doi.org/10.4230/LIPIcs.SoCG.2016.7
    Aichholzer, Oswin ; Alvarez, Victor ; Hackl, Thomas ; Pilz, Alexander ; Speckmann, Bettina ; Vogtenhuber, Birgit. / An improved lower bound on the minimum number of triangulations. 32nd International Symposium on Computational Geometry, SoCG 2016. Band 51 Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH, 2016. S. 7.1-7.16
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