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Convex Polygons in Geometric Triangulations†

Part of: Extremal combinatorics Graph theory Algorithms - Computer Science General convexity

Published online by Cambridge University Press:  30 May 2017

ADRIAN DUMITRESCU  and
CSABA D. TÓTH
ADRIAN DUMITRESCU
Affiliation:
Department of Computer Science, University of Wisconsin–Milwaukee, USA (e-mail: [email protected])
CSABA D. TÓTH
Affiliation:
Department of Mathematics, California State University Northridge, Los Angeles, CA, USA Department of Computer Science, Tufts University, Medford, MA, USA (e-mail: [email protected])
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Abstract

We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029n). This improves an earlier bound of O(1.6181n) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

MSC classification

Primary: 05C85: Graph algorithms
Secondary: 05D99: None of the above, but in this section 52A10: Convex sets in $2$ dimensions (including convex curves) 68W05: Nonnumerical algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 5 , September 2017 , pp. 641 - 659
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

†

An extended abstract of this paper appeared in the Proceedings of the 29th International Symposium on Algorithms and Data Structures (WADS 2015), Vol. 9214 of Lecture Notes in Computer Science, Springer, 2015, pp. 289–300.

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