Let G be a graph, and let H be a subgraph of G drawn in a surface Sigma. When can this drawing be extended to an embedding of the whole of G in Sigma, 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. (C) 1995 Academic press, Inc. [References: 7]
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