Graph Minors: XV. Giant Steps

LetGbe a graph with a subgraphHdrawn with high representativity on a surfaceΣ. When can the drawing ofHbe extended “up to 3-separations” to a drawing ofGinΣif we permit a bounded number (κsay) of “vortices” in the drawing ofG, that is, local areas of non-planarity? (The caseκ=0 was studied in the previous paper of this series.) For instance, if there is a path inGwith ends inH, far apart, and otherwise disjoint fromH, then no such extension exists. We are concerned with the converse; if no extension exists, what can we infer aboutG? It turns out that either there is a path as above, or one of two other obstructions is present.