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:=[C1,...,Cp,...,C2⁢p−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∩Cp⊆P∩Cp. This result applies to any graph that is partially embedded on a disk (where C is also embedded). In fact, we prove this result in the most general version where the linkage L is s-scattered: every two vertices of distinct paths are within a distance bigger than 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.