  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\in R, then there exists s\in S so that S-\{s\}\cup
\{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 (n-wheels where n\geq 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.
