Voltage graphs
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TL;DR
Although this new method is but the dual of the current graph construction, it has independent importance in practical applications, illustrated here by a computation of the genus of a class of metacyclic groups.
Abstract
A possible way to obtain a complicated graph imbedding in a surface is to derive it as a covering of a simpler imbedding by assigning "voltages" to the edges of the simpler imbedding. Although this new method is but the dual of the current graph construction,it has independent importance in practical applications, illustrated here by a computation of the genus of a class of metacyclic groups.