On rooted node-connectivity problems

Let G be a graph which is k-outconnected from a specified root node r, that is, G has k openly disjoint paths between r and v for every node v. We give necessary and sufficient conditions for the existence of a pair rv, rw of edges for which replacing these edges by a new edge vw gives a graph that is k-outconnected from r. This generalizes a theorem of Bienstock et al. on splitting off edges while preserving k-node-connectivity. We also prove that if C is a cycle in G such that each edge in C is critical with respect to k-outconnectivity from r, then C has a node v, distinct from r, which has degree k. This result is the rooted counterpart of a theorem due to Mader. We apply the above results to design approximation algorithms for the following problem: given a graph with nonnegative edge weights and node requirements c_u for each node u, find a minimum-weight subgraph that contains max(c_u, c_v) openly disjoint paths between every pair of nodes u, v. For metric weights, our approximation guarantee is 3. For uniform weights, our approximation guarantee is min[2, (k + 2q-1)/k]. Here k is the maximum node requirement, and q is the number of positive node requirements.