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Blocking the \(k\)-Holes of Point Sets in the Plane

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Abstract

Let \(P\) be a set of \(n\) points in the plane in general position. A subset \(H\) of \(P\) consisting of \(k\) elements that are the vertices of a convex polygon is called a \(k\)-hole of \(P\), if there is no element of \(P\) in the interior of its convex hull. A set \(B\) of points in the plane blocks the \(k\)-holes of \(P\) if any \(k\)-hole of \(P\) contains at least one element of \(B\) in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of \(k\)-hole blocking sets, with emphasis in the case \(k=5\).

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Notes

  1. Another proof of this result has independently been found recently by Valtr, inspired by discussions during a meeting in Spain in May 2011 (personal communication).

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Authors and Affiliations

  1. Instituto de Matemáticas, UNAM, Mexico, Mexico

    Javier Cano & Jorge Urrutia

  2. Departamento de Métodos Estadísticos, IUMA, Universidad de Zaragoza, Zaragoza, Spain

    Alfredo García & Javier Tejel

  3. Departament de Matemàtica Aplicada II, UPC, Barcelona, Spain

    Ferran Hurtado

  4. Research Institute of Educational Development, Tokai University, Tokyo, Japan

    Toshinori Sakai

Authors
  1. Javier Cano
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  2. Alfredo García
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  3. Ferran Hurtado
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  4. Toshinori Sakai
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  5. Javier Tejel
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  6. Jorge Urrutia
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Corresponding author

Correspondence to Javier Cano.

Additional information

A. García, Ferran Hurtado and J. Tejel were partially supported by projects E58 (ESF)-DGA, MICINN MTM2009-07242, MINECO MTM2012-30951, and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306 J. Urrutia was partially supported by projects MTM2006-03909 (Spain), and SEP-CONACYT of Mexico, Proyecto 80268.

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Cano, J., García, A., Hurtado, F. et al. Blocking the \(k\)-Holes of Point Sets in the Plane. Graphs and Combinatorics 31, 1271–1287 (2015). https://doi.org/10.1007/s00373-014-1488-z

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  • DOI: https://doi.org/10.1007/s00373-014-1488-z

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