Covering Islands in Plane Point Sets

  • Ruy Fabila-Monroy
  • Clemens Huemer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Let S be a set of n points in general position in the plane. A k-island I of S is a subset of k points of S such that Conv(I) ∩ S = I. We show that, for an arbitrary but fixed number k ≥ 2, the minimum number of k-islands among all sets S of n points is Θ(n2). The following related counting problem is also studied: For l < k, an l-island covers a k-island if it is contained in the k-island. Let Ck,l(S) be the minimum number of l-islands needed to cover all the k-islands of S and let Ck,l(n) be the minimum of Ck,l(S) among all sets S of n points. We show asymptotic bounds for Ck,l(n).


planar point sets islands Horton sets 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ruy Fabila-Monroy
    • 1
  • Clemens Huemer
    • 2
  1. 1.Departamento de MatemáticasCentro de Investigación y de Estudios Avanzados del Instituto Politécnico NacionalMexico
  2. 2.Departament de Matemàtica Aplicada IVUniversitat Politècnica de CatalunyaSpain

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