The Minimal Automorphism-Free Tree

TL;DR

This paper proves that within the set of all finite automorphism-free trees ordered by leaf deletion, there exists a unique minimal tree, answering a question posed by Rupinski.

Contribution

The paper establishes the existence and uniqueness of a minimal automorphism-free tree in the poset defined by leaf deletions.

Findings

There is a unique minimal automorphism-free tree.

The poset of automorphism-free trees has a least element.

The result confirms a conjecture by Rupinski.

Abstract

A finite tree $T$ with $|V(T)| \geq 2$ is called {\it automorphism-free} if there is no non-trivial automorphism of $T$. Let $\mathcal{AFT}$ be the poset with the element set of all finite automorphism-free trees (up to graph isomorphism) ordered by $T_1 \preceq T_2$ if $T_1$ can be obtained from $T_2$ by successively deleting one leaf at a time in such a way that each intermediate tree is also automorphism-free. In this paper, we prove that $\mathcal{AFT}$ has a unique minimal element. This result gives an affirmative answer to the question asked by Rupinski.