On the excluded minor structure theorem for graphs of large treewidth

TL;DR

This paper introduces a new, simplified version of the excluded minor structure theorem for graphs with large treewidth, aiming for broader applicability and easier proofs within graph minor theory.

Contribution

It presents a more accessible, unified version of the structure theorem, combining traditional and novel techniques to simplify proofs and enhance future applications.

Findings

New simplified structure theorem for graphs of large treewidth

Combines traditional and novel proof techniques

Provides properties useful for future graph minor research

Abstract

At the core of the Robertson-Seymour theory of graph minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor. Robertson and Seymour prove several versions of this theorem, each stressing some particular aspects needed at a corresponding stage of the proof of the main result of their theory, the graph minor theorem. We prove a new version of this structure theorem: one that seeks to combine maximum applicability with a minimum of technical ado, and which might serve as a canonical version for future applications in the broader field of graph minor theory. Our proof departs from a simpler version proved explicitly by Robertson and Seymour. It then uses a combination of traditional methods and new techniques to derive some of the more subtle features of other versions as well as further useful properties, with substantially simplified proofs.