AbstractA generalized type of graph covering, called a “Wrapped quasicovering” (wqc) is defined. IfK, Lare graphs dually embedded in an orientable surfaceS, then we may lift these embeddings to embeddings of dual graphs K̃,L̃ in orientable surfaces S̃, such that S̃ are branched covers ofSand the restrictions of the branched coverings to K̃,L̃ are wqc's ofK, L.the theory is applied to obtain genus embeddings of composition graphsG[nK1] from embeddings of “quotie