Bonds with parity constraints

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 vertical bar S-i boolean AND A(j)vertical bar 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 vertical bar A(1)vertical bar = vertical bar A(2)vertical bar = vertical bar A(3)vertical bar = 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. (C) 2011 Elsevier Inc. All rights reserved.