**Abstract**

We give examples to show that not even edNCE, the most general known notion of context-free graph grammar, is suited for the specification of some common data structures.

To overcome this problem, we use monadic second-order logic and
introduce *edge constraints* as a new means of specifying a
large class of graph families. Our notion stems from a natural
dichotomy found in programming practice between ordinary pointers
forming spanning trees and auxiliary pointers cutting across.

Our main result is that for certain transformations of graphs definable in monadic second-order logic, the question of whether a graph family given by a specification A is mapped to a family given by a specification B is decidable. Thus a decidable Hoare logic arises.