Enumeration of graph embeddings
Let \(G\) be a finite connected simple graph, with \(\Gamma\) a subgroup of \(\Aut G\). Two 2-cell imbeddings \(i: G\to S\) and \(j: G\to S\) of \(G\) into a closed surface \(S\) (orientable or nonorientable) are said to be congruent with respect to \(\Gamma\) if there exist a surface homeomorphism \(h: S\to S\) and a \(\gamma\in \Gamma\) such that \(h\circ i= j\circ \gamma\). The authors give an algebraic characterization of congruent 2-cell imbeddings (with respect to \(\Gamma\)). This is used to enumerate the corresponding congruence classes. This work substantially generalizes that of \textit{B. P. Mull}, \textit{R. G. Rieper} and the reviewer [Proc. Am. Math. Soc. 103, No. 1, 321-330 (1988; Zbl 0653.05028)], which deals with the oriented congruence of imbeddings of a fixed graph into orientable surfaces, with respect to \(\Gamma= \Aut G\).
scientific article; zbMATH DE number 716453
scientific article