Ribbon graph

, the edge rectangles become long and thin like ribbons, giving the name to the representation.

One may recover the surface itself by gluing a topological disk to the ribbon graph along each boundary component.

[1] 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.

[3] 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.

However, ribbon graphs are also applied in knot theory,[4] and in this application weaker notions of equivalence that take into account the 3d embedding may also be used.

A ribbon graph with one vertex (the yellow disk), three edges (two of them twisted), and one face. It represents an embedding of a graph with three self-loops onto the connected sum of three projective planes .