In this paper we describe a Curry-like type system for graphs and extend it with uniqueness information to indicate that certain objects are only `locally accessible'. The correctness of type assignment guarantees that no external access on such an object will take place in the future. We prove that types are preserved under reduction (for both type systems) for a large class of rewrite systems. Adding uniqueness information provides a solution to two problems in implementations of functional languages: efficient space management and interfacing with non-functional operations.