Generating closed 2-cell embeddings in the torus and the projective plane (Q1819729)

If a graph is embedded in a 2-manifold such that all faces are cells bounded by simple closed curves, it is called a closed 2-cell embedding of the graph in the manifold. Among the various results for the recursive generation of 2-cell complexes and their graphs the Steinitz-theorem on planar 3-connected graphs is by far the best known. The author proves that the 2-cell embeddings in the projective plane can be generated from two minimal graphs, and in the case of the torus from six minimal graphs by vertex splitting and face splitting.