Non-embeddable extensions of embedded minors

A graph G is weakly 4-connected if it is 3-connected, has at least five vertices, and for every pair (A,B) such that A∪B=V(G), |A∩B|=3 and no edge has one end in A−B and the other in B−A, one of the induced subgraphs G[A],G[B] has at most four edges. We describe a set of constructions that starting from a weakly 4-connected planar graph G produce a finite list of non-planar weakly 4-connected graphs, each having a minor isomorphic to G, such that every non-planar weakly 4-connected graph H that has a minor isomorphic to G has a minor isomorphic to one of the graphs in the list. Our main result is more general and applies in particular to polyhedral embeddings in any surface.