I am studying imbeddings of (connected, undirected, unweighted, multi-)graphs on oriented surfaces of arbitrary genus, and I am searching for a reference for the statement that the dual of the dual graph is the original graph. I have looked at several references, but I have not found a statement that is suitably general for my purposes (and even proofs of the statement are surprisingly rare). In Mohar and Thomassen's *Graphs on Surfaces*, the authors define graph imbeddings purely combinatorially, using rotation systems, and they do not prove that the dual of the dual of a graph is the original graph, merely stating that this is "clear". In Gross and Tucker's *Topological Graph Theory*, the authors offer a definition of the dual graph which I do not believe is sufficiently precise to avoid cases where the double dual is not the original graph. Nevertheless they assert that the double dual is the original graph, again without proof.

I have been considering graph imbeddings from a geometric perspective, where the edges are imbedded in my surface as simple curves, and where two edge imbeddings intersect only possibly at common endpoints. Thus, my definition differs from the combinatorial approach, but it offers an increased flexibility: the combinatorial definition will only yield faces that are all 2-cells, where I would like to be able to consider more general imbeddings. In any case, it seems like some sort of double-duality statement is true for these general imbeddings.

Can you point me to a reference which considers graph imbeddings from this geometric perspective, and that actually proves that the double-dual is the original graph? (As a bonus: can someone explain to me whether "imbedding" or "embedding" is the more widely accepted spelling?)