# On Colourability of Polygon Visibility Graphs

## TL;DR

This paper investigates the complexity of colouring visibility graphs of polygons, providing polynomial algorithms for 4-colouring simple polygons and proving NP-completeness for 5-colouring and polygons with holes.

## Contribution

It introduces polynomial algorithms for 4-colouring simple polygons and establishes NP-completeness results for 5-colouring and polygons with holes.

## Key findings

- Polynomial-time algorithm for 4-colouring simple polygons
- NP-completeness of 5-colouring simple polygons
- NP-completeness of 4-colouring polygons with holes

## Abstract

We study the problem of colouring visibility graphs of polygons. In particular, for visibility graphs of simple polygons, we provide a polynomial algorithm for 4-colouring, and prove that the 5-colourability question is already NP-complete for them. For visibility graphs of polygons with holes, we prove that the 4-colourability question is NP-complete.