Selecting vertex disjoint paths in plane graphs

We study variants of the vertex disjoint paths problem in plane graphs where paths have to be selected from given sets of paths. We investigate the problem as a decision, maximization, and routing-in-rounds problem. Although all considered variants are np-hard in planar graphs, restrictions on the locations of the terminals on the outer face of the given planar embedding of the graph lead to polynomially solvable cases for the decision and maximization versions of the problem. For the routing-in-rounds problem, we obtain a p-approximation algorithm, where p is the maximum number of alternative paths for a terminal pair, when restricting the locations of the terminals to the outer face such that they appear in a counterclockwise traversal of the boundary as a sequence for some permutation.