Coloring vertices of a graph or finding a Meyniel obstruction

TL;DR

This paper presents efficient algorithms to find either a clique and coloring or a Meyniel obstruction in a graph, advancing graph coloring and recognition techniques.

Contribution

It introduces O(n^2) and O(n^3) algorithms for detecting Meyniel obstructions and related structures in graphs, improving computational methods in graph theory.

Findings

O(n^2) algorithm for clique and coloring detection

O(n^3) algorithm for strong stable set detection

Efficient recognition of Meyniel obstructions

Abstract

A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds either a clique and coloring of the same size or a Meyniel obstruction. We also give a O(n^3) algorithm that, for any graph, finds either aneasily recognizable strong stable set or a Meyniel obstruction.