Seymour's conjecture on 2-connected graphs of large pathwidth

TL;DR

This paper proves Seymour's conjecture that large pathwidth 2-connected graphs necessarily contain certain apex-forest or outerplanar minors, advancing understanding of graph minors and structure.

Contribution

It provides a proof of Seymour's conjecture on minors in 2-connected graphs with large pathwidth, confirming a long-standing hypothesis in graph theory.

Findings

Confirmed Seymour's conjecture for apex-forest and outerplanar minors

Established a bound on pathwidth for containing specific minors

Contributed to the theory of graph minors and structural graph theory

Abstract

We prove the conjecture of Seymour (1993) that for every apex-forest $H_1$ and outerplanar graph $H_2$ there is an integer $p$ such that every 2-connected graph of pathwidth at least $p$ contains $H_1$ or $H_2$ as a minor. An independent proof was recently obtained by Dang and Thomas.