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Data exchange and schema mapping management have received little attention so far in the graph database scenario, and tools developed in this context for relational databases have significant drawbacks in the context of graph-structured data. In this paper we embark on the study of interoperability issues for graph databases, including schema mappings, data exchange and certain answers computation. We start by analyzing different possibilities for specifying mappings in graph databases. Our mapping languages are based on the most typical graph databases queries, ranging from regular path queries to conjunctions of nested regular expressions. They subsume all previously considered mapping languages, and let one express many data exchange scenarios in the graph database context. We study the problems of materializing solutions and query answering, in particular, the problem of computing universal representatives and certain answers for various classes of mappings. We show that both problems are difficult with respect to combined complexity, and that for the latter problem, even data complexity is high for some very simple mappings and queries. We then identify relevant classes of mappings and queries for which the problems of materializing solutions and query answering can be solved efficiently.