Abstract
A matroid has the max-flow min-cut property if a certain circuit packing problem has an optimal solution that is integral whenever the capacities assigned to the elements of the matroid are integral. P.D. Seymour characterized the matroids with this property in terms of minimal forbidden minors. Here we employ a variant of a previously developed decomposition algorithm to produce two decomposition theorems for this matroid class. The first theorem roughly says that any 3-connected matroid of the class is regular, or equal to the Fano matroid, or is a 3-sum. The second theorem is quite similar, but involves a more detailed analysis of the 3-sum case and includes an additional case.
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