Minors of two-connected graphs of large path-width

TL;DR

This paper proves that large path-width 2-connected graphs necessarily contain certain minors, specifically graphs P or Q, answering longstanding questions and advancing understanding of graph minors.

Contribution

It establishes a bound on path-width ensuring the presence of specific minors in 2-connected graphs, using a novel property of tree-decompositions.

Findings

Existence of a bound p for minors P or Q in large path-width graphs

Answer to Seymour's question about minors in 2-connected graphs

Confirmation of Marshall and Wood's conjecture

Abstract

Let $P$ be a graph with a vertex $v$ such that $P\backslash v$ is a forest, and let $Q$ be an outerplanar graph. We prove that there exists a number $p=p(P,Q)$ such that every 2-connected graph of path-width at least $p$ has a minor isomorphic to $P$ or $Q$. This result answers a question of Seymour and implies a conjecture of Marshall and Wood. The proof is based on a new property of tree-decompositions.