Abstract
In 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the equivalence class of a tree under the embeddability relation is, up to isomorphism, either trivial or infinite. In 2022 Abdi, et al. provided a rigorous exposition of a counter-example to TAC developed by Tetano in his 2008 PhD thesis. In this paper we provide a positive answer to TAC for a weaker type of graph relation: the topological minor relation. More precisely, letting [T] denote the equivalence class of T under the topological minor relation we show that1.|[T]|=1 or |[T]|≥ℵ0 and2.∀r∈V(T), |[(T,r)]|=1 or |[(T,r)]|≥ℵ0. In particular, by means of curtailing trees, we show that for any tree T with at least one ray with infinitely many vertices with degree at least 3: |[T]|≥2ℵ0.
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