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Given a graph G and k pairs of vertices (s_1,t_1), ..., (s_k,t_k), thek-Vertex-Disjoint Paths problem asks for pairwise vertex-disjoint paths P_1,..., P_k such that P_i goes from s_i to t_i. Schrijver [SICOMP'94] proved thatthe k-Vertex-Disjoint Paths problem on planar directed graphs can be solved intime n^{O(k)}. We give an algorithm with running time 2^{2^{O(k^2)}} n^{O(1)}for the problem, that is, we show the fixed-parameter tractability of theproblem.

The planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable