  Consider the following problem: given a graph with edge costs and a subset Q
of vertices, find a minimum-cost subgraph in which there are two edge-disjoint
paths connecting every pair of vertices in Q. The problem is a
failure-resilient analog of the Steiner tree problem arising, for example, in
telecommunications applications. We study a more general mixed-connectivity
formulation, also employed in telecommunications optimization. Given a number
(or requirement) r(v) in {0, 1, 2} for each vertex v in the graph, find a
minimum-cost subgraph in which there are min{r(u), r(v)} edge-disjoint u-to-v
paths for every pair u, v of vertices.
  We address the problem in planar graphs, considering a popular relaxation in
which the solution is allowed to use multiple copies of the input-graph edges
(paying separately for each copy). The problem is max SNP-hard in general
graphs and strongly NP-hard in planar graphs. We give the first polynomial-time
approximation scheme in planar graphs. The running time is O(n log n).
  Under the additional restriction that the requirements are only non-zero for
vertices on the boundary of a single face of a planar graph, we give a
polynomial-time algorithm to find the optimal solution.
