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.

A set S of vertices 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 G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that ifm is the prime factor decomposition of a connected graph then Det(G) = max{Det(G k i i )}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Q n ) = log 2 n +1 which matches the lower bound, and that Det(K n 3 ) = log 3 (2n+1) +1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(H n ) = (log n).᭧

This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels. A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertex set using $1, \ldots, d$ so that no nontrivial automorphism of $G$ preserves the labels. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. We prove that a graph is $d$-distinguishable if and only if it has a determining set that can be $(d-1)$-distinguished. We use this to prove that every Kneser graph $K_{n:k}$ with $n\geq 6$ and $k\geq 2$ is $2$-distinguishable.

A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. Call the minimum size of a label class in such a labeling of G the cost of 2-distinguishing G and denote it by ρ(G). This paper shows that for n ≥ 5, log 2 n + 1 ≤ ρ(Q n ) ≤ 2 log 2 n − 1, where Q n is the hypercube of dimension n.

A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U . A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W . Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and r ∈ R, then there exists s ∈ S so that S − {s} ∪ {r} is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchange property. This paper shows that neither determining sets nor resolving sets have the exchange property in all graphs, but that both have the exchange property in trees. It also gives an infinite graph family (nwheels where n ≥ 8) in which determining sets have the exchange property but resolving sets do not. Further, this paper provides necessary and sufficient conditions for determining sets to have the exchange property in an outerplanar graph.