I know that any graph isomorphism defines an equivalence relation on said graph. But now I have some equivalence relation on a graph. Is it possible to construct an isomorphism? Intuitively I would say yes maybe even multiple ones (I am not sure how many...) since I can map elements in each equivalence class to elements in the same equivalence class in such a way that their neighborhood relationships are preserved.
But is their a canonical formal way of writing down the corresponding isomorphism?
Thank you for your help!