Group action for enumerating maps on surfaces (Q1429348)

A map is a 2-cell imbedding of a connected pseudograph \(G\) into a surface, which can be either orientable or nonorientable. The authors introduce the concept of the semi-arc automorphism group of \(G\) to classify all imbeddings of \(G\) under the action of this group. They enumerate the rooted maps on orientable and nonorientable surfaces, with underlying graph \(G\). By this new method, many closed formulas are re-established, such as for complete and complete bipartite graphs, bouquets, dipoles, and generalized dipoles.