Generalizing the duality theorem of graph embeddings

Let ω: Ḡ→G be a wrapped covering of graphs and let G be enbedded in a surface S. It is known that if the embedding of G in S is orientable, then this embedding may be lifted to an orientable embedding of Ḡ in S̄. In this paper, we generalize this result by showing that the embedding of G in S may be lifted to an embedding of Ḡ in S̄, where S̄ has the same orientability characteristic as that of S. Furthermore, ω extends to a branched covering B:S̄→S of surfaces and the restriction of B to the dual of Ḡ is a wrapped covering onto the dual of G. These results were applied to obtain genus embeddings of composition graphs G[K̄_n] from embeddings of the graph G.