We show that for pairs (Q,R) and (S,T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V(G)−(Q∪R∪S∪T) such that there are two ways to reduce G by a vertex-minor operation while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids. Joint work with Sang-il Oum.
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