On better-quasi-ordering under graph minors

TL;DR

The paper links finite graphs’ BQO status to the WQO/BQO status of countable rayless graphs under graph minors, establishing a web of equivalent conditions that connect rank-based decompositions, minor-twin classification, and transfinite induction. It introduces and leverages marked graphs, suspensions, and a robust rank framework to translate infinite-graph phenomena into finite-graph analogues, enabling novel equivalences and reductions. Key results include Seymour’s self-minor conjecture for rayless graphs, a Borel-criterion for minor-closed forest families, and a transfinite BWQO program showing that finitary BQO implies the WQO/BQO status of broader classes. The work provides tools and a roadmap for tackling Thomas’ conjectures in the rayless regime and suggests pathways to attack planarity and tree-width variants via marked/unmarked techniques and rank-based induction.

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 $α$ can be decomposed into exactly $\aleph_0$ minor-twin classes for every ordinal $α<ω_1$. 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.

Figures (4)

• Figure 1: The components of $G-A$ and $G- (A-v)$ in the proof of \ref{['KG']}.
• Figure 2: The vertex set $A":= A(H) \cup \bigcup_{Q\in \mathcal{S}'} A(Q)$ in the proof of \ref{['lt beta']}, enclosed by the dashed curve (blue).
• Figure 3: The graph $G'_j$ in the proof of \ref{['MeqA']} (middle), produced by joining copies of $G_j$ (left), and an attempt to embed it into $G'_{j+1}$ (right).
• Figure 4: Defining $B'_{u_i}$ in the proof of \ref{['CinCp']}. The left picture depicts a portion of $G$, while the right picture depicts its minor model in $G'$. The dotted curves enclose the original branch sets, while the dashed curve (in blue) encloses $B'_{u_2}$.

Theorems & Definitions (89)

• Conjecture 1.1: Folklore DiestelBook25PeqTow
• Conjecture 1.2: Thomas' conjecture ThoWel
• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1: DiestelBook25
• Theorem 2.3: GM23
• Lemma 2.5
• proof