Graphs and Combinatorics

, Volume 23, Issue 5, pp 481–507 | Cite as

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles

  • O. Aichholzer
  • C. Huemer
  • S. Kappes
  • B. Speckmann
  • Cs. D. Tóth
Article

Abstract

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.

Keywords

Partitions decompositions planar point sets Ramsey-type results 

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Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

  • O. Aichholzer
    • 1
  • C. Huemer
    • 2
  • S. Kappes
    • 3
  • B. Speckmann
    • 4
  • Cs. D. Tóth
    • 5
  1. 1.Institute for Software TechnologyGraz University of TechnologyAustria
  2. 2.Departament de Matemática Aplicada IIUniv. Poli. de CatalunyaSpain
  3. 3.Department of Mathematics TU BerlinGermany
  4. 4.Department of Mathematics and Computer Science TU EindhovenNetherland
  5. 5.Department of MathematicsMITCambridegeUSA

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