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 $d:V\to \mathbb{Z}$, 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 $w:E\to (0,\infty)$ 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.
Consider an undirected graph G = (V, E). A subgraph of G is a subset of its edges, whilst an orientation of G is an assignment of a direction to each edge. Provided with an integer circulationdemand 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 [5] showed, using dimensional methods, that the number of subgraphs k-connecting a vertex s to a vertex t is the same as the number of orientations k-connecting s to t. An application of our result is an efficient, bijective proof of this fact.
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