Studying graph theory I found the following theorem for s,t vertex of the graph $G=(V,E)$ and a cut defined as "subset $C$ of $E$ an s−t cut if C = $δ ^{out}(U)$ for some subset U of V satisfying s $\in$ U and t $\notin$ U:
The minimum length of an s − t path is equal to the maximum number of disjoint s − t cuts.
Can someone prove it to me?
Let $P$ be a shortest $s-t$ path.
Then on the one hand, every cut $C$ has to contain at least one edge $e$ in $P$ [make sure you see why]. Thus in any family $\{C_1,\ldots, C_m\}$ such that the $C_j$s are pairwise disjoint, it follows that $m \le |E(P)|$ lest there is an edge in $P$ in at least two of the $C_j$s (which would contradict pairwise disjointness).
On the other hand, for each $j=1,2,\ldots, |E(P)|$ let $C_j$ be the set of edges in $G$ cutting the vertices of precisely distance $j-1$ from $s$ from the vertices of distance precisely $j$ from $s$. Then the $C_j$s are disjoint and each is an $s-t$ cut [make sure you see why].