A new min-max theorem concerning bi-supermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Gy} ori's theorem on intervals, W. Mader's theorem on splitting o edges in directed graphs, J. Edmonds' theorem on matroid partitions, and an earlier result of the rst author on the minimum number of new directed edges whose addition makes a digraph k-edge-connected.As another consequence, we solve the corresponding node-connectivity augmentation problem in directed graphs.
By applying the matroid partition theorem of J. Edmonds [1] to a hypergraphic generalization of graphic matroids, due to M. Lorea [3], we obtain a generalization of Tutte's disjoint trees theorem for hypergraphs. As a corollary, we prove for positive integers k and q that every (kq)-edge-connected hypergraph of rank q can be decomposed into k connected sub-hypergraphs, a well-known result for q = 2. Another by-product is a connectivity-type sufficient condition for the existence of k edge-disjoint Steiner trees in a bipartite graph.
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