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We investigate the question of when two entity-relationship diagrams (ERD's) should be considered equivalent, in the sense of representing the same information. This question is very important for a database design process which uses the ERD model, and can be interpreted in various ways. We give three natural and increasingly stricter criteria for developing concepts of equivalence for ERD's. We first give a notion of "domain data compatibility" which ensures that the ERD's in question represent the same universe of data in an aggregate sense. Then we define the set of functional dependencies which are naturally embedded in each ERD, and use it to develop a concept of "data dependency equivalence" which ensures that the ERD's satisfy the same constraints (functional dependencies) among the represented data. Finally, we give our strongest criterion, instance data equivalence, which requires the ERD's to have the same power to represent instances of data. We develop several alternate forms of this third notion, including some giving efficient tableaux tests for its occurrence. Indeed, for each type of equivalence, we give a polynomial-time algorithm to test for it.