Distances and cuts in planar graphs

We prove the following theorem. Let G = (V, E) be a planar bipartite graph, embedded in the euclidean plane. Let O and I be two of its faces. Then there exist pairwise edge-disjoint cuts C1, …, Ct so that for each two vertices v, w with v, w ϵ O of v, w ϵ I, the distance from v to w in G is equal to the number of cuts Cj separating v and w. This theorem is dual to a theorem of Okamura on plane multicommodity flows, in the same way as a theorem of Karzanov is dual to one of Lomonosov.

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