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Most Finite Point Sets in the Plane have Dilation \(>1\)

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Abstract

We prove the following fact for arbitrary finite point sets \(S\) in the plane. Either, S is a subset of one of the well-known sets of points whose triangulation is unique and has dilation 1. Or there exists a number \(\Delta (S) > 1\) such that each finite plane graph containing \(S\) among its vertices has dilation \(\ge \Delta (S)\).

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Acknowledgments

This work was partially supported by DFG (Kl 655/ 14-1/3).

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Authors and Affiliations

  1. Department of Computer Science I, University of Bonn, Bonn, Germany

    Rolf Klein, Martin Kutz & Rainer Penninger

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  1. Rolf Klein
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  2. Martin Kutz
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  3. Rainer Penninger
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Corresponding author

Correspondence to Rainer Penninger.

Additional information

M. Kutz—deceased 2007.

An extended abstract of this paper has appeared at SoCG’06 [16].

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Klein, R., Kutz, M. & Penninger, R. Most Finite Point Sets in the Plane have Dilation \(>1\) . Discrete Comput Geom 53, 80–106 (2015). https://doi.org/10.1007/s00454-014-9651-0

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  • DOI: https://doi.org/10.1007/s00454-014-9651-0

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