This paper initiates a study on the problem of computing the difference between the metric dimension and the determining number of graphs. We provide new proofs and results on the determining number of trees and Cartesian products of graphs, and establish some lower bounds on the difference between the two parameters. Let G be a connected graph1 . A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number is the smallest size of a determining set...