Isomorphisms of Graph Embeddings

Question

I am having trouble understanding isomorphisms of graph embeddings. How does one distinguish between two graph embeddings on the same surface, and how does one, for example, distinguish between two embeddings $G\to S$ and $G'\to S'$? When do we consider these embeddings "the same"?

Answer 1

It depends. It depends on what you're doing, and which embeddings you want to consider the same.

The most natural thing I can think of would be to say two graph embeddings $f \colon G \to S$ and $g\colon G' \to S$ are "the same" if there exists a graph isomorphism $\phi\colon G \to G'$ and an automorphism $\alpha$ of your surface $S$ such that $g\phi = \alpha f$. But you could very well weaken the conditions on $\phi$ and $\alpha$.