Generalizing the duality theorem of graph embeddings (Q1263592)

A wrapped covering \(\omega: \bar G\to G\) of a connected graph \(\bar G\) onto a connected graph G is a continuous surjection mapping vertices onto vertices and arcs onto arcs preserving the preferred direction and satisfying: (i) there is a positive integer N such that \(| \omega^{-1}(e)| =N\) for each edge e of G, and (ii) for vertex \(\bar v\) of \(\bar G,\) there is a positive integer \(\delta(\bar v)\) such that, if \(v=\omega (\bar v)\), then for each edge e of G directed into (out of) v, there are \(\delta(\bar v)\) edges of \(\bar G\) in \(\omega^{-1}(e)\) directed into (out of) \(\bar v.\) The theory of wrapped coverings was introduced by \textit{T. D. Parsons}, \textit{T. Pisanski}, and \textit{B. Jackson} [Discrete Math. 31, 43-52 (1980; Zbl 0453.05023), and J. Graph Theory 5, 55-77 (1981; Zbl 0423.05012)]. In the present paper the duality theorem of the second reference above (for orientable imbeddings) is generalized as follows: Let \(\omega: \bar G\to G\) be a wrapped covering of graphs, with G imbedded in a closed 2- manifold S (either orientable or nonorientable). This imbedding can be lifted to an imbedding of \(\bar G\) in \(\bar S,\) where \(\bar S\) is a closed 2-manifold of the same orientability characteristic as that of S. Moreover, \(\omega\) extends to a branched covering \(B: \bar S\to S\) of surfaces and the restriction of B to the dual of \(\bar G\) is a wrapped covering onto the dual of G. Several applications are given, such as the construction of triangular imbeddings of compositions \(G[\bar K_ n]\) from certain triangular imbeddings of G. In this connection, see also \textit{A. Bouchet} [J. Graph Theory 6, 57-74 (1982; Zbl 0488.05032)].