Computational Aspects of Problems on Visibility and Disk Graph Representations

TL;DR

This thesis explores computational problems related to visibility and disk graphs, providing new algorithms, complexity results, and models that better reflect real-world limitations in geometric graph representations.

Contribution

It introduces polynomial algorithms for certain visibility problems, proves NP-completeness of coloring problems, and models realistic constraints with unit disk visibility graphs and recognition complexity results.

Findings

Polynomial-time algorithm for conflict-free polygon guarding.

NP-completeness of 5-colorability in simple polygon visibility graphs.

NP-hardness of recognizing axes-parallel unit disk graphs.

Abstract

This thesis focuses on two concepts which are widely studied in the field of computational geometry. Namely, visibility and unit disk graphs. In the field of visibility, we have studied the conflict-free chromatic guarding of polygons, for which we have described a polynomial-time algorithm that uses $O(n \log^2 n)$ colors to guard a polygon in a conflict-free setting, and proper coloring of polygon visibility graphs, for which we have described an algorithm that returns a proper 4-coloring for a simple polygon. Besides, we have shown that the 5-colorability problem is NP-complete on visibility graphs of simple polygons, and 4-colorability is NP-complete on visibility graphs of polygons with holes. Then, we move further with the notion of visibility, and define a graph class which considers the real-world limitations for the applications of visibility graphs. That is, no physical object has infinite range, and two objects might not be mutually visible from a certain distance although there are no obstacles in-between. To model this property, we introduce unit disk visibility graphs, and show that the 3-colorability problem is NP-complete for unit disk visibility graphs of a set of line segments, and a polygon with holes. After bridging the gap between the visibility and the unit disk graphs, we then present our results on the recognition of unit disk graphs in a restricted setting -- axes-parallel unit disk graphs. We show that the recognition of unit disk graphs is NP-complete when the disks are centered on pre-given parallel lines. If, on the other hand, the lines are not parallel to one another, the recognition problem is NP-hard even though the pre-given lines are axes-parallel (i.e. any pair is either parallel or perpendicular).