Identifying Graph Automorphisms Using Determining Sets

Abstract

A set of vertices $S$ is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of a graph is the size of a smallest determining set. This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to investigate determining sets. Further, determining sets of Kneser graphs are extensively studied, sharp bounds for their determining numbers are provided, and all Kneser graphs with determining number $2$, $3,$ or $4$ are given.