Visibility-preserving convexifications using single-vertex moves

Ábrego, Bernardo M.; Cetina, M.; Leaños, Jesús; Salazar, Gelasio

doi:10.1016/j.ipl.2011.11.012

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Convexifying Monotone Polygons while Maintaining Internal Visibility

Aichholzer

Cetina

Fabila-Monroy

et al. 2012

Lecture Notes in Computer Science

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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 [7,8]: (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.

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