Visibility in Graphs and Polygons

In this thesis, we investigate visibility in connection with path planning. We focus on two problems concerning the exploration of and the orientation in an environment. The first problem we consider is the Watchperson Route Problem , in which mobile agents are deployed to monitor the interior of a 2-dimensional environment bounded by a polygon. The goal is to see (a) every part of, or (b) some points of interest in the environment at some point in time. To save resources, the route along which a mobile agent walks shall be as short as possible. We explore different variations of this problem, involving human vision as we know it, as well as a notion of vision that resembles that of a modem with the ability of seeing through some walls. Our research is driven by the question of finding an optimal solution―a shortest path―with efficient use of resources. As it turns out, some variations of the Watchperson Route Problem allow for computing optimal solutions with only short computation time, while other variations are provably difficult to solve optimally. For those variations that are hard to solve, we provide efficient algorithms to approximate optimal solutions. In the second problem, we are given an odd number of points in the plane, and non-intersecting segments such that each segment connects two of the points, leaving only one point unmatched. We ask whether it is possible to reconfigure this arrangement into any other arrangement of equally many non-intersecting segments on the given point set by gradually adding and removing one segment in such a way that after each reconfiguration step, the number of segments remains constant and only one point is unmatched. To keep this transformation planar, the segments are viewed as obstacles in the plane, and the visibility between the unmatched point and the segment endpoints is used to determine possible transformations. We prove that there always exists a sequence of reconfiguration steps that allows us to transform any given set of segments into any other set of segments on the same set of endpoints.