# Min-Cost-Flow Preserving Bijection Between Subgraphs and Orientations

### Abstract

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.