The classical Whitney's Theorem in low topological theory/graph theory states that every 3-connected planar graph is uniquely embeddable (up to orientation) on the sphere.

My question is the following: let's consider an small genus surface (for instance, the projective plane or the thorus with only 1 hole). This result is already not true, but I was wondering which extra conditions one may ask to the graph we want to embed to assure uniqueness on the embedding.

Of course here we are asking about "cellular embeddings" (all faces defined by the embedding are contractible)

I know a result due to Robertson and Vitray concerning representability of the graph, but I would like to know if there is more results on this direction (concerning, for instance, vertex degrees, minimum level of connectivity, etc)

The classical Whitney Theorem in low topological theory/graph theory states that every 3-connected planar graph is uniquely embeddable (up to orientation) on the sphere.

My question is the following: let's consider an small genus surface (for instance, the projective plane or the torus with only 1 hole). This result is already not true, but I was wondering which extra conditions one may ask of the graph we want to embed to assure uniqueness on the embedding.

Of course here we are asking about "cellular embeddings" (all faces defined by the embedding are contractible)

I know a result due to Robertson and Vitray concerning representability of the graph, but I would like to know if there are more results in this direction (concerning, for instance, vertex degrees, minimum level of connectivity, etc.)