We consider mixed networks, which may include both directed and undirected edges. For a nontrivial vertex subset S, an S-disconnecting set is a set of edges and vertices which intersects every path from any vertex in S to any vertex not in S. Given nonnegative edge and vertex costs, we show that the minimum cost of an S-disconnecting set defines a submodular function. This implies that the set of all S inducing minimum-cost disconnecting sets is the set of closures of a binary relation, thus extending Picard-Queyranne's result on ordinary minimum cuts. We apply this result to two-pair multicommodity problems in undirected networks, extending Hu's result to disconnecting sets that may include vertices as well as edges. These results and a result of Provan and Shier may be used for generating all sets S that induce such minimum-cost disconnecting sets and ranking such sets in order of corresponding costs, for both one-pair problems in mixed networks and two-pair problems in undirected networks.
