A shorter proof of the path-width theorem

TL;DR

This paper presents a more concise and simpler proof of a theorem relating path-width of graphs to the containment of all forests of a certain size as minors.

Contribution

It provides a shorter, more straightforward proof of a known path-width theorem, improving upon previous proofs by Diestel.

Findings

Shorter proof of the path-width theorem

Simplifies understanding of graph minors related to path-width

Reduces proof complexity for the theorem

Abstract

A graph has {\em path-width} at most $w$ if it can be built from a sequence of graphs each with at most $w+1$ vertices, by overlapping consecutive terms. Every graph with path-width at least $w-1$ contains every $w$-vertex forest as a minor: this was originally proved by Bienstock, Robertson, Thomas and the author, and was given a short proof by Diestel. Here we give a proof even shorter and simpler than that of Diestel.