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We introduce a two-player nonlocal game, called the (G,H)-isomorphism game, where classical players can win with certainty if and only if the graphs G and H are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the (G,H)-isomorphism game, respectively. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. We prove that non-signalling isomorphismarXiv:1611.09837v3 fatcat:vayzru5tgvb57f4wrr45vwqonq