Argumentation is a dynamic process. The enforcing problem in argumentation, i.e. the question whether it is possible to modify a given argumentation framework (AF) in such a way that a desired set of arguments becomes an extension or a subset of an extension, was first studied in [3] and positively answered under certain conditions. In this paper, we take up this research and study the more general problem of minimal change. That is, in brief, i) is it possible to enforce a desired set of arguments, and if so, ii) what is the minimal number of modifications (additions or removals of attacks) to reach such an enforcement, the so-called characteristic. We show for several Dung semantics that this problem can be decided by local criteria encoded by the so-called value functions. Furthermore, we introduce the corresponding equivalence notions between two AFs which guarantee equal minimal efforts needed to enforce certain subsets, namely minimal-E-equivalence and the more general minimal change equivalence. We present characterization theorems for several Dung semantics and finally, we show the relations to standard and the recently proposed strong equivalence [9] for a whole range of semantics.