The Cycling Property for the Clutter of Odd st-Walks

A binary clutter is cycling if its packing and covering linear program have integral optimal solutions for all eulerian edge capacities. We prove that the clutter of odd st-walks of a signed graph is cycling if and only if it does not contain as a minor the clutter of odd circuits of K 5 nor the clutter of lines of the Fano matroid. Corollaries of this result include, of many, the characterization for weakly bipartite signed graphs [5], packing two-commodity paths[7,10], packing T-joins with small |T|, a new result on covering odd circuits of a signed graph, as well as a new result on covering odd circuits and odd T-joins of a signed graft.

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