Let s1 , t1 ,. . . sk , tk be vertices in a graph G embedded on a surface Σ of genus g. A vertex v of G is "redundant" if there exist k vertex disjoint paths linking si and ti (1 ≤ i ≤ k) in G if and only if such paths also exist in G − v. Robertson and Seymour proved in Graph Minors VII that if v is "far" from the vertices si and tj and v is surrounded in a planar part of Σ by l(g, k) disjoint cycles, then v is redundant. Unfortunately, their proof of the existence of l(g, k) is not constructive. In this paper, we give an explicit single exponential bound in g and k.
