On better-quasi-ordering under graph minors

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 $\hat{I}\pm$ can be decomposed into exactly ${\aleph }_0$ minor-twin classes for every ordinal $\hat{I}\pm <\ddot{I}{‰}_1$. 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.