Transformations in closed 2-cell embeddings on surfaces preserving specified properties (Q1292865)

Let \({\mathcal C}={\mathcal C}(S)\) be a class of 2-cell embedded graphs (possibly with multiple edges) in a surface \(S\), such that each face is bounded by a cycle, and with the property that each embedded graph with some minor in \({\mathcal C}\) also belongs to \({\mathcal C}\). The author proves that for any surface \(S\) there exists a number \(n_{\mathcal C}(S)\) such that any two embedded graphs \(G,G'\in {\mathcal C}\) with \(|E(G)|=|E(G')|\geq n_{\mathcal C}(S)\) are related by a sequence of certain local transformations.