Graph minors. VI. Disjoint paths across a disc

Let G be a graph drawn on a disc, and let the vertices of G on the boundary of the disc be s₁, …, s_k, t₁, …, t_k in some order. When are there k vertex-disjoint paths of G joining s_i and t_i (1 ≤ i ≤ k), respectively? We give a structural characterization and a polynomial algorithm for this problem. We also solve the same question when the disc is replaced by a cylinder.