Components of branched coverings of current graphs

Abstract Desired graph imbeddings are obtained as branched coverings of simpler imbeddings, continuing earlier work of the authors. This paper inaugurates a systematic approach to the problem of devising appropriate current assignments. Chief results include the establishment of a bijective correspondence between the components of the cover and the cosets of the isotropy group of an arbitrary component, leading to a proof that the components are mutually isomorphic regular branched coverings of the dual of the current graph imbedding. Also given is a characterization of the isotropy group via a face labeling technique that aids in the construction of imbeddings.