Fixing Numbers of Graphs and Groups

TL;DR

This paper characterizes the fixing sets of finite abelian groups and explores the fixing sets of symmetric groups, contributing to understanding automorphism fixing parameters in graph theory.

Contribution

It provides a complete characterization of fixing sets for finite abelian groups and investigates fixing sets for symmetric groups, expanding knowledge on automorphism fixing parameters.

Findings

Fixing sets of finite abelian groups are fully characterized.

Fixing sets of symmetric groups are analyzed and partially characterized.

The fixing number relates to the distinguishing number as a variation with unique labels.

Abstract

The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex in $S$. The fixing set of a group $\Gamma$ is the set of all fixing numbers of finite graphs with automorphism group $\Gamma$. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label $G$ so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.