A simple proof for visibility paths in simple polygons Mohammad Reza Zarrabi a , Nasrollah Moghaddam Charkari a,* a Faculty of Electrical Engineering and Computer Science, Tarbiat Modares University, Tehran, Iran Abstract The purpose of this note is to give a simple proof for a necessary and sufficient condition for visibility paths in simple polygons. A visibility path is a curve such that every point inside a simple polygon is visible from at least one point on the path. This result is essential for finding the shortest watchman route inside a simple polygon specially when the route is restricted to curved paths. Keywords: Computational Geometry; Visibility Paths; Watchman Route; Simple Polygons; 1. Introduction Visibility coverage of simple polygons with a mobile guard (mainly known as watchman problems) has been central problems in computational geometry. Usually, a mobile guard is defined as a moving point inside a simple polygon P that sees in any direction for any distance. Also, visibility is defined by the clearance of straight line between two points inside P . In other words, two points inside P see each other if the line segment connecting them does not intersect the boundary of P . Consider the curved path created by the mobile guard during its walk inside P . Coverage is achieved if every point inside P is visible from at least one point on the path. In this case, the path is called a visibility path. The watchman route problem (WRP) asks for finding a visibility route (a visibility path whose starting and end points coincide) inside P of minimum length. The WRP is defined for two versions. The anchored WRP, in which the tour is required to pass through a specified anchor point, and the floating WRP, in which no anchor point is specified. The problem has a polynomial time solution for both versions. The anchored WRP was first studied by Chin and Ntafos [2]. Afterwards, many researchers have worked on this problem in [12], [11], [5], [13] and [4], respectively, to improve the results. The floating WRP was investigated in [1, 4]. The best known results for both versions were presented by Dror et al. [4]. Linear time algorithms are known for approximating the anchored WRP [9] and floating WRP [10] (also, a non-linear time algorithm in [8]). All the above solutions for the WRP used the following theorem (special case of the theorem for visibility routes): * Corresponding author Email addresses: m.zarabi@modares.ac.ir (Mohammad Reza Zarrabi), charkari@modares.ac.ir (Nasrollah Moghaddam Charkari) Preprint submitted to arXiv November 23, 2021 arXiv:2004.02227v3 [cs.CG] 15 Jul 2020