Disjoint paths in tournaments

Abstract Given k pairs of vertices ( s i , t i ) ( 1 ≤ i ≤ k ) of a digraph G, how can we test whether there exist k vertex-disjoint directed paths from s i to t i for 1 ≤ i ≤ k ? This is NP-complete in general digraphs, even for k = 2 [2] , but for k = 2 there is a polynomial-time algorithm when G is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen [1] . Here we prove that for all fixed k there is a polynomial-time algorithm to solve the problem when G is semicomplete.