Distances and cuts in planar graphs

The paper proves 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∈O or v,w∈I, the distance from v to w in G is equal to the number of cuts Ct 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 Lomonsov.