Automorphism Groups and Adversarial Vertex Deletions

TL;DR

This paper introduces a game-theoretic framework where a player constructs graphs with specific automorphism groups, responding to an adversary’s challenges involving vertex deletions that alter the automorphism group.

Contribution

It generalizes previous constructions by enabling a player to respond to an adversary's sequence of group challenges through a flexible graph construction.

Findings

The player can always respond to any sequence of adversarial group challenges.

The construction encodes multiple finite groups as automorphism groups of graphs and their vertex-deleted subgraphs.

The framework extends the understanding of automorphism groups in graph theory.

Abstract

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the automorphism group of a graph and a vertex-deleted subgraph. In this note, we describe a generalized scenario as a game between a player and an adversary: An adversary provides a list of finite groups and a number of rounds. The player constructs a graph with automorphism group isomorphic to the first group. In the following rounds, the adversary selects a group and the player deletes a vertex such that the automorphism group of the corresponding vertex-deleted subgraph is isomorphic to the selected group. We provide a construction that allows the player to appropriately respond to any sequence of challenges from the adversary.