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Given a connected graph G=(V, E) and three even-sized subsets A 1, A 2, A 3 of V, when does V have a partition (S 1, S 2) such that G[S i] is connected and |S i∩A j| is odd for all i=1, 2 and j=1, 2, 3? This problem arises in the area of integer flow theory and has theoretical interest in its own right. The special case when |A 1|=|A 2|=|A 3|=2 has been resolved by Chakravarti and Robertson, and the general problem can be rephrased as a problem on binary matroids that asks if a given triple of elements is contained in a circuit. The purpose of this paper is to present a complete solution to this problem based on a strengthening of Seymour's theorem on triples in matroid circuits. © 2011 Elsevier Inc.

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Journal of Combinatorial Theory. Series B

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