On fractional multicommodity flows and distance functions

The authors give some results on the existence of fractional and integral solutions to multicommodity flow problems, and on the related problem of decomposing distance functions into cuts. One of the results is: Let G=(V,E) be a planar bipartite graph. Then there exists subsets W1,...,Wt of V so that for each pair v', v“ of vertices on the boundary of G, the distance of v' and v” in G is equal to the number of j=1,...,t with ℝ{v',v“}ℝWjℝ=1 and so that the cuts δ(Wj) are pairwise disjoint.