Preselecting Homotopies for the Weighted Disjoint Paths Problem

The work of Robertson and Seymour implies that the disjoint paths problem is polynomial solvable for a fixed number of terminals. Even, Itai and Shamir showed that a weighted version of the problem is NP-hard even for a demand graph consisting of two edges. In the present paper, it is shown that the weighted disjoint paths problem is polynomial solvable for graphs embedded on a fixed surface and fixed demand graph. An alternative formulation of the problem is: given a graph G embedded in a plane with a certain number of forbidden regions, find a maximum weight collection of pairwise internally disjoint paths connecting nodes as specified by some fixed demand graph. The results imply that this problem is polynomial solvable if the number of forbidden regions is bounded. One consequence is the existence of a polynomial-time algorithm for finding a maximum collection of induced paths between specified nodes in a graph embedded on a fixed surface.