The Visibility Freeze-Tag Problem

In the Freeze-Tag Problem, we are given a set of robots at points inside some metric space. Initially, all the robots are frozen except one. That robot can awaken (or “unfreeze”) another robot by moving to its position, and once a robot is awakened, it can move and help to awaken other robots. The goal is to awaken all the robots in the shortest time. The Freeze-Tag Problem has been studied in different metric spaces: graphs and Euclidean spaces. In this thesis, we look at the Freeze-Tag Problem in polygons, and we introduce the Visibility Freeze-Tag Problem, where one robot can awaken another robot by “seeing” it. Furthermore, we introduce a variant of the Visibility Freeze-Tag Problem, called the Line/Point Freeze Tag Problem, where each robot lies on an awakening line, and one robot can awaken another robot by touching its awakening line. We survey the current results for the Freeze-Tag Problem in graphs, Euclidean spaces and polygons. Since the Visibility Freeze-Tag Problem bears some resemblance to the Watchman Route Problem, we also survey the background literature on the Watchman Route Problem. We show that the Freeze-Tag Problem in polygons and the Visibility Freeze-Tag Problem are NP-hard, and we present an O(n)-approximation algorithm for the Visibility Freeze-Tag Problem. For the Line/Point Freeze-Tag Problem, we give a polynomial time algorithm for the special case where all the awakening lines are parallel to each other. We prove that the general case is NP-hard, and we present an O(1)- approximation algorithm

Visibility graphs and landscape visibility analysis

Visibility analysis based on viewsheds is one of the most frequently used GIS analysis tools. In this paper we present an approach to visibility analysis based on the visibility graph. A visibility graph records the pattern of mutual visibility relations in a landscape, and provides a convenient way of storing and further analysing the results of multiple viewshed analyses for a particular landscape region. We describe how a visibility graph may be calculated for a landscape. We then give examples, which include the interactive exploration ofa landscape, and the calculation of new measures of a landscape?s visual properties based on graph metrics ? in particular, neighbourhood clustering coefficient and path length analysis. These analyses suggest that measures derived from the visibility graph may be of particular relevance to the growing interest in quantifying the perceptual characteristics of landscapes

A Characterization of Visibility Graphs for Pseudo-Polygons

In this paper, we give a characterization of the visibility graphs of pseudo-polygons. We first identify some key combinatorial properties of pseudo-polygons, and we then give a set of five necessary conditions based off our identified properties. We then prove that these necessary conditions are also sufficient via a reduction to a characterization of vertex-edge visibility graphs given by O'Rourke and Streinu

Transforming planar graph drawings while maintaining height

There are numerous styles of planar graph drawings, notably straight-line drawings, poly-line drawings, orthogonal graph drawings and visibility representations. In this note, we show that many of these drawings can be transformed from one style to another without changing the height of the drawing. We then give some applications of these transformations

On Visibility Representations of Non-planar Graphs

A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge