Edge-disjoint homotopic paths in a planar graph with one hole

We prove the following theorem, conjectured by K. Mehlhorn: Let G = (V, E) be a planar graph, embedded in the plane C. Let O denote the interior of the unbounded face, and let I be the interior of some fixed bounded face. Let C1, …, Ck be curves in Cs(I⌣O), with end points in V⌢bd(I⌣O), so that for each vertex v of G the degree of v in G has the same parity as the number of curves Ci beginning or ending in v (counting a curve beginning and ending in v for two). Then there exist pairwise edge-disjoint paths P1, …, Pk in G so that Pi is homotopic to Ci in the space Cs(I⌣O) for i = 1, …, k, if and only if for each dual walk Q from {I, O} to {I, O} the number of edges in Q is not smaller than the number of times Q necessarily intersects the curves Ci. The theorem generalizes a theorem of Okamura and Seymour. We demonstrate how a polynomial-time algorithm finding the paths can be derived.