Approximate minimum cuts and their enumeration

We show that every $\alpha$-approximate minimum cut in a connected graph is the unique minimum $(S,T)$-terminal cut for some subsets $S$ and $T$ of vertices each of size at most $\lfloor 2\alpha \rfloor +1$. This leads to an alternative proof that the number of $\alpha$-approximate minimum cuts in a $n$-vertex connected graph is $n^{O(\alpha )}$ and they can all be enumerated in deterministic polynomial time for constant $\alpha$.