Combing a Linkage in an Annulus [article]

Petr A. Golovach and Giannos Stamoulis and Dimitrios M. Thilikos
2022 arXiv   pre-print
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
more » ... n 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,...,C_2p-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∩ C_p⊆ P∩ C_p. 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.
arXiv:2207.04798v1 fatcat:arquew3onfcu5ff5ql7iicppwu