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Holes in 2-Convex Point Sets

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Combinatorial Algorithms (IWOCA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10765))

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Abstract

Let S be a set of n points in the plane in general position (no three points from S are collinear). For a positive integer k, a k-hole in S is a convex polygon with k vertices from S and no points of S in its interior. For a positive integer l, a simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. A point set S is l-convex if there exists an l-convex polygonization of S.

Considering a typical Erdős–Szekeres-type problem, we show that every 2-convex point set of size n contains an \(\varOmega (\log n)\)-hole. In comparison, it is well known that there exist arbitrarily large point sets in general position with no 7-hole. Further, we show that our bound is tight by constructing 2-convex point sets with holes of size at most \(O(\log n)\).

Research supported by OEAD project CZ 18/2015 and by project no. 7AMB15A T023 of the Ministry of Education of the Czech Republic. O. Aichholzer and B. Vogtenhuber supported by ESF EUROCORES programme Euro-GIGA - ComPoSe, Austrian Science Fund (FWF): I648-N18. M. Balko and P. Valtr supported by grant GAUK 690214, by project CE-ITI no. P202/12/G061 of the Czech Science Foundation GAČR, and by ERC Advanced Research Grant no 267165 (DISCONV). T. Hackl supported by Austrian Science Fund (FWF): P23629-N18. A. Pilz supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35. P. Ramos supported by MINECO project MTM2014-54207, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, for Spain.

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References

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Acknowledgments

This work was initiated during the ComPoSe Workshop on Algorithms using the Point Set Order Type held in March/April 2014 in Ratsch, Austria.

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Authors and Affiliations

  1. Institute for Software Technology, Graz University of Technology, Graz, Austria

    Oswin Aichholzer, Thomas Hackl & Birgit Vogtenhuber

  2. Department of Applied Mathematics and Institute for Theoretical Computer Science (CE-ITI), Charles University, Prague, Czech Republic

    Martin Balko & Pavel Valtr

  3. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Martin Balko

  4. Department of Computer Science, ETH Zürich, Zürich, Switzerland

    Alexander Pilz

  5. Departamento de Física y Matemáticas, Universidad de Alcalá, Alcalá de Henares, Spain

    Pedro Ramos

Authors
  1. Oswin Aichholzer
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  2. Martin Balko
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  3. Thomas Hackl
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  4. Alexander Pilz
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  5. Pedro Ramos
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  6. Pavel Valtr
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  7. Birgit Vogtenhuber
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Corresponding author

Correspondence to Martin Balko .

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Editors and Affiliations

  1. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Ljiljana Brankovic

  2. University of Newcastle Australia, Callaghan, New South Wales, Australia

    Joe Ryan

  3. McMaster University, Hamilton, Ontario, Canada

    William F. Smyth

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Aichholzer, O. et al. (2018). Holes in 2-Convex Point Sets. In: Brankovic, L., Ryan, J., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2017. Lecture Notes in Computer Science(), vol 10765. Springer, Cham. https://doi.org/10.1007/978-3-319-78825-8_14

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  • DOI: https://doi.org/10.1007/978-3-319-78825-8_14

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  • Print ISBN: 978-3-319-78824-1

  • Online ISBN: 978-3-319-78825-8

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