A unified algorithm for colouring graphs of bounded clique-width

TL;DR

This paper introduces a unified algorithm for graph colouring based on clique-width, efficiently handling both small and large numbers of colours, and improving speed in many cases.

Contribution

It presents a novel unified algorithm that combines two extreme-case algorithms for clique-width graph colouring, achieving optimal complexity in both scenarios.

Findings

Achieves state-of-the-art complexity for both small and large colour counts.

Provides a speed-up for colouring graphs with many colours.

Unifies previously separate algorithms into a single, simpler approach.

Abstract

Clique-width is one of the graph complexity measures leading to polynomial special-case algorithms for generally NP-complete problems, e.g. graph colourability. The best two currently known algorithms for verifying c-colourability of graphs represented as clique-width terms are optimised towards two different extreme cases, a constant number of colours and a very large number of colours. We present a way to unify these approaches in a single relatively simple algorithm that achieves the state of the art complexity in both cases. The unified algorithm also provides a speed-up for a large number of colours.