Minor exclusion in quasi-transitive graphs

TL;DR

This paper establishes a characterization of locally finite quasi-transitive graphs that are quasi-isometric to trees, linking their minor exclusion property to the quasi-isometry class, thus generalizing previous results on groups.

Contribution

It introduces a new equivalence between being quasi-isometric to a tree and minor exclusion for locally finite quasi-transitive graphs, extending prior group theory results.

Findings

Locally finite quasi-transitive graphs quasi-isometric to trees are characterized by minor exclusion.

Minor exclusion is equivalent to being quasi-isometric to a tree within this class.

The result generalizes earlier theorems from group theory to graph theory.

Abstract

In this note, we show that locally finite quasi-transitive graphs are quasi-isometric to trees if and only if every other locally finite quasi-transitive graph quasi-isometric to them is minor excluded. This generalizes results by Ostrovskii and Rosenthal and by Khukhro on minor exclusion for groups.