Automorphism groups of graphs of bounded Hadwiger number

TL;DR

This paper characterizes the automorphism groups of finite graphs with bounded Hadwiger number, revealing they are constructed through specific group extensions involving abelian, symmetric, and bounded order groups.

Contribution

It provides a structural description of automorphism groups for graphs excluding large minors, especially for edge-transitive, twin-free graphs, using group extension techniques.

Findings

Automorphism groups are built from abelian, symmetric, and bounded order groups.

Non-abelian composition factors have bounded order in certain graphs.

Automorphism groups are obtained via repeated group extensions.

Abstract

We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. Our proof includes a structural analysis of finite edge-transitive graphs. In particular, we show that for connected, $K_{h+1}$-minor-free, edge-transitive, twin-free, finite graphs the non-abelian composition factors of the automorphism group have bounded order. We use this to show that the automorphism groups of finite graphs of bounded Hadwiger number are obtained by repeated group extensions using abelian groups, symmetric groups and groups of bounded order.