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. Received September 15, 1998; revised March 10, 2000, and April 17, 2000.