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Graphs and Combinatorics

, Volume 31, Issue 5, pp 1271–1287 | Cite as

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

  • Javier Cano
  • Alfredo García
  • Ferran Hurtado
  • Toshinori Sakai
  • Javier Tejel
  • Jorge Urrutia
Original Paper

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 \(k=5\).

Keywords

\(k\)-holes Piercing Blocking 

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Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Javier Cano
    • 1
  • Alfredo García
    • 2
  • Ferran Hurtado
    • 3
  • Toshinori Sakai
    • 4
  • Javier Tejel
    • 2
  • Jorge Urrutia
    • 1
  1. 1.Instituto de Matemáticas, UNAMMexicoMexico
  2. 2.Departamento de Métodos Estadísticos, IUMAUniversidad de ZaragozaZaragozaSpain
  3. 3.Departament de Matemàtica Aplicada IIUPCBarcelonaSpain
  4. 4.Research Institute of Educational DevelopmentTokai UniversityTokyoJapan

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