On better-quasi-ordering under graph minors

TL;DR

This paper explores the complex relationship between better-quasi-ordering and graph minors, proving several equivalences and implications for specific classes of graphs, including rayless and tree graphs, and addresses longstanding open problems.

Contribution

It establishes new equivalences between well-quasi-ordering and better-quasi-ordering for graphs with no infinite paths, and proves Seymour's self-minor conjecture for rayless graphs.

Findings

Countable graphs with no infinite paths are equivalent to finite graphs being better-quasi-ordered.

Rayless countable graphs of any rank can be decomposed into countably many minor-twin classes.

A minor-closed family of N-labelled rayless forests is Borel iff it excludes all rayless forests.

Abstract

In the aftermath of the Robertson--Seymour Graph Minor Theorem, Thomas conjectured that the countable graphs are well-quasi-ordered under the minor relation. We prove that this conjecture, when restricted to graphs with no infinite paths (rays), is equivalent to the statement that the finite graphs are better-quasi-ordered, another well-known open problem. Even more, we prove that the latter implies that the countable rayless graphs are better-quasi-ordered. We prove several other statements to be equivalent to the above, one of which being that the rayless countable graphs of rank $\alpha$ can be decomposed into exactly $\aleph_0$ minor-twin classes for every ordinal $\alpha<\omega_1$. By restricting the latter statement to trees, and combining it with Nash-Williams' theorem that the infinite trees are well-quasi-ordered, we deduce as a side result that a minor-closed family of N-labelled rayless forests is Borel -- in the Tychonoff product topology -- if and only if it does not contain all rayless forests. As another side-result, we prove Seymour's self-minor conjecture for rayless graphs of any cardinality.