Two ribbon graph representations are said to be equivalent (and define homeomorphic graph embeddings) if they are related to each other that a homeomorphism of the topological space formed by the union of the vertex disks and edge rectangles that preserves the identification of these features. Ribbon graph representations may be equivalent even if it is not possible to deform one into the other within 3d space: this notion of equivalence considers only the intrinsic topology of the representation, and not how it is embedded.
