Abstract
Extending theorems of Rado and Lovász, we introduce a new framework for problems concerning supermodular functions and graphs. Among the application is an optimization problem for finding a minimum-cost subgraph H of a digraph G=( V, E) such that H contains k disjoint paths from a fixed paths from a node of G to any other node. Another consequence is a characterization for graphs having a branching that meets all directed cuts. A theorem of Vidyasankar on optimal covering by arborescences and a matroid intersection theorem of Gröflin and Hoffman are also shown to be special cases.
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