Extensions of Signed Graphs
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Given a signed graph (G, Σ) with an embedding on a surface S, we are interested in "extending" (G, Σ) by adding edges and splitting vertices, such that the resulting graph has no embedding on S. We show (assuming 3-connectivity for (G, Σ)) that there are a small number of minimal extensions of (G, Σ) with no such embedding, and describe them explicitly. We also give conditions, for several surfaces S, for an embedding of a signed graph on S to extend uniquely. These results find application in characterizing the signed graphs with no odd-K_5 minor.