Role colouring graphs in hereditary classes

TL;DR

This paper investigates the computational complexity of role colouring in hereditary graph classes, identifying boundary classes that separate polynomial-time solvable cases from NP-hard ones, and introduces a novel technique for generating regular graphs.

Contribution

It provides the first boundary class for k-role colouring and related problems, advancing understanding of complexity boundaries in hereditary graph classes.

Findings

Identified a boundary class for k-role colouring.

Developed a technique for generating regular graphs of arbitrary girth.

Established complexity distinctions within hereditary classes.

Abstract

We study the computational complexity of computing role colourings of graphs in hereditary classes. We are interested in describing the family of hereditary classes on which a role colouring with k colours can be computed in polynomial time. In particular, we wish to describe the boundary between the "hard" and "easy" classes. The notion of a boundary class has been introduced by Alekseev in order to study such boundaries. Our main results are a boundary class for the k-role colouring problem and the related k-coupon colouring problem which has recently received a lot of attention in the literature. The latter result makes use of a technique for generating regular graphs of arbitrary girth which may be of independent interest.