Min-Cost-Flow Preserving Bijection Between Subgraphs and Orientations

Consider an undirected graph G = (V, E). A subgraph of G is a subset of its edges, while an orientation of G is an assignment of a direction to each of its edges. Provided with an integer circulation–demand d: V → Z, we show an explicit and efficiently computable bijection between subgraphs of G on which a d-flow exists and orientations on which a d-flow exists. Moreover, given a cost function w: E → (0, ∞) we can find such a bijection which preserves the w-min-cost-flow. In 2013, Kozma and Moran [Electron. J. Comb. 20(3)] showed, using dimensional methods, that the number of subgraphs k-edge-connecting a vertex s to a vertex t is the same as the number of orientations k-edge-connecting s to t. An application of our result is an efficient, bijective proof of this fact.