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Counting Carambolas

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Abstract

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of n points in the plane. Configurations of interest include convex polygons, star-shaped polygons and monotone paths. We also consider related problems for directed planar straight-line graphs.

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Notes

  1. Throughout this paper, all logarithms are in base 2.

  2. Such paths were inadvertently overlooked in the analysis from [13].

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Acknowledgments

A. Dumitrescu was supported in part by NSF grant DMS-1001667. M. Löffler was supported by the Netherlands Organization for Scientific Research (NWO) under grant 639.021.123. Research by Tóth was supported in part by NSERC (RGPIN 35586) and NSF (CCF-1423615). This work was initiated at the workshop “Counting and Enumerating Plane Graphs,” which took place at Schloss Dagstuhl in March, 2013.

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Authors and Affiliations

  1. Department of Computer Science, University of Wisconsin-Milwaukee, Milwaukee, USA

    Adrian Dumitrescu

  2. Department of Computing and Information Sciences, Utrecht University, Utrecht, The Netherlands

    Maarten Löffler

  3. LG Theoretische Informatik, FernUniversität Hagen, Hagen, Germany

    André Schulz

  4. Department of Mathematics, California State University Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

  5. Department of Computer Science, Tufts University, Medford, MA, USA

    Csaba D. Tóth

Authors
  1. Adrian Dumitrescu
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  2. Maarten Löffler
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  3. André Schulz
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  4. Csaba D. Tóth
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Corresponding author

Correspondence to Csaba D. Tóth.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 25th Canadian Conference on Computational Geometry [13].

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Dumitrescu, A., Löffler, M., Schulz, A. et al. Counting Carambolas. Graphs and Combinatorics 32, 923–942 (2016). https://doi.org/10.1007/s00373-015-1621-7

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  • DOI: https://doi.org/10.1007/s00373-015-1621-7

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