Enumeration of graph embeddings Original Research Article

For a finite connected simple graph G, let Γ be a group of graph automorphisms of G. Two 2-cell embeddings ι: G → S and j: G → S of a graph G into a closed surface S (orientable or nonorientable) are congruent with respect to Γ if there are a surface homeomorphism h:S → S and a graph automorphism γϵΓ such that hoι=joγ. In this paper, we give an algebraic characterization of congruent 2-cell embeddings, from which we enumerate the congruence classes of 2-cell embeddings of a graph G into closed surfaces with respect to a group of automorphisms of G, not just the full automorphism group. Some applications to complete graphs are also discussed. As an orientable case, the oriented congruence of a graph G into orientable surfaces with respect to the full automorphism group of G was enumerated by Mull et al. (1988).