Graph Minors: XV. Giant Steps

Abstract Let G be a graph with a subgraph H drawn with high representativity on a surface Σ . When can the drawing of H be extended “up to 3-separations” to a drawing of G in Σ if we permit a bounded number ( κ say) of “vortices” in the drawing of G , that is, local areas of non-planarity? (The case κ =0 was studied in the previous paper of this series.) For instance, if there is a path in G with ends in H , far apart, and otherwise disjoint from H , then no such extension exists. We are concerned with the converse; if no extension exists, what can we infer about G ? It turns out that either there is a path as above, or one of two other obstructions is present.