A Proof of the Tree Alternative Conjecture Under the Topological Minor Relation

TL;DR

This paper proves the Tree Alternative Conjecture for the topological minor relation, establishing conditions under which trees have unique or infinitely many topological minor equivalence classes, with implications for curtailing trees.

Contribution

It extends the Tree Alternative Conjecture to the topological minor relation, providing new results on the size of equivalence classes of trees under this relation.

Findings

Equivalence classes of trees are either singleton or infinite under topological minor relation.

Trees with at least one non-eventually bare ray have continuum many equivalence classes.

The paper introduces methods for curtailing trees to analyze their topological minor classes.

Abstract

We prove the Tree Alternative Conjecture for the topological minor relation: letting $[T]$ denote the equivalence class of $T$ under the topological minor relation we show that: $|[T]| = 1$ or $|[T]|\geq \aleph_0$ and $\forall r\in V(T)$, $|[(T,r)]| = 1$ or $|[(T,r)]|\geq \aleph_0$. In particular, by means of curtailing trees, we show that for any tree $T$ with at least one not eventually bare ray: $|[T]| \geq 2^{\aleph_0}$.