Transformations in closed 2-cell embeddings on surfaces preserving specified properties Original Research Article

Consider a class C of closed 2-cell embeddings on a closed surface F2, allowed to have multiple edges, such that for an embedding G ϵ C, each embedding which has G as a minor is also an element of C. In this paper, we shall show that for any closed surface F2, there exists a natural number NC(F2) such that any two elements G, G′ ϵ C with |E(G)β = |E(G′)| ⩾ NC(F2) are connected in C by a sequence of the local transformations.