On Essential and Inessential Polygons in Embedded Graphs

In this article, we present a number of results of the following type: A given subgraph of an embedded graph either is embedded in a disc or it contains a face chain that has a non-contractible curve through it. One application is that any two faces of a 4representative embedding are simultaneously contained in a disc bounded by a polygon. Such results are applied to giving proofs of the existence of a non-contractible separating polygon and of the existence of large numbers of pairwise disjoint non-contractible polygons. Some of these latter results were previously obtained by other authors.