EGC 2011: Computational Geometry pp 98-108

# 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)

## Abstract

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.

## Keywords

convexification monotone polygons visibility graph

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## 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