The graph minor relation satisfies the twin alternative conjecture

TL;DR

This paper proves that the Tree Alternative Conjecture holds for the graph minor relation, showing that the equivalence class of a graph under minors is either trivial or infinite.

Contribution

It extends the validity of the Tree Alternative Conjecture to the graph minor relation, completing the understanding for the three major graph relations.

Findings

TAC holds for the graph minor relation.

Completes the classification for embeddability, topological minor, and graph minor.

Advances the theory of graph relations and their equivalence classes.

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 LaFlamme, et al. provided a rigorous exposition of a conter-example to TAC developed by Tetano in his 2008 PhD thesis. Also in 2022, the present author provided a positive answer to TAC for the topological minor relation. Along with embeddability and the topological minor, the graph minor relation completes the triad of the most widely studied graph relations. In this paper we provide a positive answer to TAC for the the graph minor.