Combing a linkage in an annulus

A linkage in a graph G of size k is a subgraph L of G whose connected components are k paths. The pattern of a linkage of size k is the set of k pairs formed by the endpoints of these paths. A consequence of the Unique Linkage Theorem is the following: there exists a function f : N -> N such that if a plane graph G contains a sequence C of at least f(k) nested cycles and a linkage of size at most k whose pattern vertices lay outside the outer cycle of C, then G contains a linkage with the same pattern avoiding the inner cycle of C. In this paper we prove the following variant of this result: Assume that all the cycles in C are "orthogonally" traversed by a linkage P and L is a linkage whose pattern vertices may lay either outside the outer cycle or inside the inner cycle of C := [C-1, . . . , C-p, . . . , C2p-1]. We prove that there are two functions g, f : N -> N, such that if L has size at most k, P has size at least f(k), and |C| >= g(k), then there is a linkage with the same pattern as L that is "internally combed" by P, in the sense that L boolean AND Cp subset of P boolean AND Cp. In fact, we prove this result in the most general version where the linkage L is s-scattered: no two vertices of distinct paths of L are within distance at most s. We deduce several variants of this result in the cases where s = 0 and s > 0. These variants permit the application of the unique linkage theorem on several path routing problems on embedded graphs.