On essential and inessential polygons in embedded graphs (Q1850578)

The authors present results of the type: A given subgraph of a graph imbedded in a surface is either imbedded in a disc or has a face chain containing a non-contractible closed path. One application is to show that any two faces of a 4-representative imbedding are simultaneously contained in a disc bounded by a polygon, and in turn to establish the existence, in an \(r\)-representative orientable imbedding, of \(\lfloor(r- 1)/8\rfloor\) pairwise disjoint, pairwise homotopic non-contractible separating polygons.