Vertex elimination orderings for hereditary graph classes

TL;DR

This paper introduces a unified method combining known tools to efficiently find elimination orderings in various hereditary graph classes, enabling improved algorithms for coloring and clique problems.

Contribution

It presents a general approach to prove and compute elimination orderings in multiple graph classes, including new classes defined by Truemper configurations.

Findings

Linear-time algorithms for elimination orderings in several graph classes

Bounds on chromatic number for studied classes

Fast algorithms for maximum clique and coloring problems

Abstract

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem.