A short proof that every finite graph has a tree-decomposition displaying its tangles

TL;DR

This paper presents a concise proof that every finite graph has a tree-decomposition displaying all maximal tangles, a key result in graph theory with extensions to matroids.

Contribution

It provides a simplified proof of a fundamental theorem linking tree-decompositions and tangles in graphs and matroids.

Findings

Every finite graph has a tree-decomposition displaying all maximal tangles.

The proof simplifies understanding of the graph minors project results.

Extension of the theorem to matroids is included.

Abstract

We give a short proof that every finite graph (or matroid) has a tree-decomposition that displays all maximal tangles. This theorem for graphs is a central result of the graph minors project of Robertson and Seymour and the extension to matroids is due to Geelen, Gerards and Whittle.