Two-Connected Augmentation Problems in Planar Graphs

Given a weighted undirected graph G and a subgraph S of G, we consider the problem of adding a minimum-weight set of edges of G to S so that the resulting subgraph satisfies specified (edge or vertex) connectivity requirements between pairs of nodes of S. This has important applications in upgrading telecommunication networks to be invulnerable to link or node failures. We give a polynomial algorithm for this problem when S is connected, nodes are required to be at most 2-connected, and G is planar. Applications to network design and multicommodity cut problems are also discussed.