Packings of cuts realizing distances between certain vertices in a planar graph

AbstractRecently A. Schrijver proved the following theorem. Suppose that G=(V, E) is a connected planar graph embedded in the euclidean plane, that O and I are two of its faces, and that the edges e∈E have nonnegative integer-valued lengths l(e) such that the length of each circuit in G is even. Then there exist cuts B1,…, Bk in G weighted by nonnegative integer-valued weights λ1,…,λk so that: (i) for each eϵE, the sum of the weights of the cuts containing e does not exceed l(e), and (ii) for each two vertices s and t both in the boundary of O or in the boundary of I, the sum of the weights of the cuts ‘separatingh’ s and t is equal to the distance between s and t.We given another proof of this theorem which provides a strongly polynomial-time algorithm for finding such cuts and weights