On colouring point visibility graphs

TL;DR

This paper investigates the computational complexity of coloring point visibility graphs, providing polynomial-time algorithms for 4-colorability, NP-completeness results for 5-colorability, and examples of graphs with specific chromatic properties.

Contribution

It establishes the polynomial-time decidability of 4-colorability and NP-completeness of 5-colorability for point visibility graphs, and presents graphs with unusual chromatic characteristics.

Findings

Deciding 4-colorability is polynomial-time solvable.

Deciding 5-colorability is NP-complete.

Existence of point visibility graphs with chromatic number 6 and clique number 4.

Abstract

In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4.