In the undirected rooted k-edge connected subgraph problem, the goal is to find a minimum cost subgraph in which there are k edge-disjoint paths between the root and each vertex in the graph. The generalized version for undirected graphs aims to find a subgraph in which each pair of vertices has k-edge disjoint paths.
Is it possible that given the optimal solution to one problem, we could create an optimal solution for the other one? I have tried to see if it is possible to construct an optimal solution for the generalized version, given the optimal solution to the rooted version. We have k paths from each node to the root, so it is possible to match these subpaths to create k paths between any pairs. However, I haven't been able to prove that such paths would necessarily be disjoint.