Unit Disk Visibility Graphs

TL;DR

This paper introduces and analyzes unit disk visibility graphs, a new class modeling visibility with distance constraints, and proves that 3-coloring is NP-complete for these graphs in certain geometric scenarios.

Contribution

It defines and classifies unit disk visibility graphs and establishes NP-completeness of 3-coloring for these graphs in specific geometric configurations.

Findings

Unit disk visibility graphs incorporate distance and obstacle considerations.

3-coloring problem is NP-complete for these graphs with line segments and polygons with holes.

Provides a new framework for more realistic visibility modeling.

Abstract

We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too far apart, regardless of having an obstacle between them. This particular graph class models real world scenarios more accurately compared to the conventional visibility graphs. We first define and classify the unit disk visibility graphs, and then show that the 3-coloring problem is NP-complete when unit disk visibility model is used for a set of line segments (which applies to a set of points) and for a polygon with holes.