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In the well-solved edge-connectivity augmentation problem we must find a minimum cardinality set F of edges to add to a given undirected graph to make it k-edge-connected. This paper solves the generalization where every edge of F must go between two different sets of a given partition of the vertex set. A special case of this partition-constrained problem, previously unsolved, is increasing the edge-connectivity of a bipartite graph to k while preserving bipartiteness. Based on this special case we present an application of our results in statics. Our solution to the general partition-constrained problem gives a min-max formula for |F| which includes as a special case the original min-max formula of Cai and Sun [Networks, 19 (1989), pp. 151--172] for the problem without partition constraints.

When k is even the min-max formula for the partition-constrained problem is a natural generalization of the unconstrained version …