Résumé
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 samepattern avoiding the inner cycle of $C$. In this paper we prove thefollowing variant of this result: Assume that all the cycles in $C$ are``orthogonally'' traversed by a linkage P and L is a linkage whosepattern vertices may lay either outside the outer cycle or inside theinner 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 samepattern as L that is ``internally combed'' by $P$ , in the sense that $L$ ∩$C_p$ ⊆ $P$ ∩ $C_p$. 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 thelinkage $L$ is s-scattered: every two vertices of distinct paths arewithin distance bigger than s. We deduce several variants of this resultin the cases where s = 0 and s > 0. These variants permitthe application of the unique linkage theorem on several path routingproblems on embedded graphs.