Graph Miners .XIV. Extending an Embedding

Let G be a graph, and let H be a subgraph of G drawn in a surface Σ. When can this drawing be extended to an embedding of the whole of G in Σ, up to 3-separations? We show that if such an extension is impossible, and if H is a subdivision of a simple 3-connected graph and is highly "representative", then one of two obstructions is present. This is a lemma for use in a future paper.