Convexifying Monotone Polygons while Maintaining Internal Visibility

  • Oswin Aichholzer
  • Mario Cetina
  • Ruy Fabila-Monroy
  • Jesús Leaños
  • Gelasio Salazar
  • Jorge Urrutia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Let P be a simple polygon on the plane. Two vertices of P are visible if the open line segment joining them is contained in the interior of P. In this paper we study the following questions posed in [8,9]: (1) Is it true that every non-convex simple polygon has a vertex that can be continuously moved such that during the process no vertex-vertex visibility is lost and some vertex-vertex visibility is gained? (2) Can every simple polygon be convexified by continuously moving only one vertex at a time without losing any internal vertex-vertex visibility during the process?

We provide a counterexample to (1). We note that our counterexample uses a monotone polygon. We also show that question (2) has a positive answer for monotone polygons.


convexification monotone polygons visibility graph 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Mario Cetina
    • 2
  • Ruy Fabila-Monroy
    • 3
  • Jesús Leaños
    • 4
  • Gelasio Salazar
    • 2
  • Jorge Urrutia
    • 5
  1. 1.Institute for Software TechnologyUniversity of TechnologyGrazAustria
  2. 2.Instituto de FísicaUniversidad Autónoma de San Luis PotosíMéxico
  3. 3.Departamento de MatemáticasCinvestavMéxico
  4. 4.Unidad Académica de MatemáticasUniversidad Autónoma de ZacatecasMéxico
  5. 5.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico

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