Automorphism groups of a graph and a vertex-deleted subgraph

TL;DR

This paper investigates which pairs of groups can be realized as automorphism groups of a graph and its vertex- or edge-deleted subgraphs, providing concrete constructions to answer these questions positively.

Contribution

It introduces a method to construct graphs with prescribed automorphism groups for the graph and its subgraphs, addressing longstanding questions in graph theory.

Findings

Pairs of groups can be realized as automorphism groups of a graph and its subgraphs.

Concrete constructions demonstrate the realizability of these group pairs.

The results extend to both vertex- and edge-deleted subgraphs.

Abstract

Understanding the structure of a graph along with the structure of its subgraphs is important for several problems in graph theory. Two examples are the Reconstruction Conjecture and isomorph-free generation. This paper raises the question of which pairs of groups can be represented as the automorphism groups of a graph and a vertex-deleted subgraph. This, and more surprisingly the analogous question for edge-deleted subgraphs, are answered in the most positive sense using concrete constructions.