Finding k partially disjoint paths in a directed planar graph

The {\it partially disjoint paths problem} is: {\it given:} a directed graph, vertices $r_1,s_1,\ldots,r_k,s_k$, and a set $F$ of pairs $\{i,j\}$ from $\{1,\ldots,k\}$, {\it find:} for each $i=1,\ldots,k$ a directed $r_i-s_i$ path $P_i$ such that if $\{i,j\}\in F$ then $P_i$ and $P_j$ are disjoint. We show that for fixed $k$, this problem is solvable in polynomial time if the directed graph is planar. More generally, the problem is solvable in polynomial time for directed graphs embedded on a fixed compact surface. Moreover, one may specify for each edge a subset of $\{1,\ldots,k\}$ prescribing which of the $r_i-s_i$ paths are allowed to traverse this edge.

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