Graph Minors: XVII. Taming a Vortex

The main result of this series serves to reduce several problems about general graphs to problems about graphs which can “almost” be drawn in surfaces of bounded genus. In applications of the theorem we usually need to encode such a nearly embedded graph as a hypergraph which can be drawn completely in the surface. The purpose of this paper is to show how to “tidy up” near-embeddings to facilitate the encoding procedure.