Finding the Min Cut

After computing max flow, the minimum cut is found using the residual graph. Run BFS or DFS from $s$ in the residual graph to find all reachable vertices. These vertices can still receive flow from $s$. Let $S$ be the set of reachable vertices and $T$ be the rest.

The cut $(S, T)$ is a minimum $s$-$t$ cut. This partition naturally separates the graph. Edges from $S$ to $T$ in the original graph are saturated (flow equals capacity). Removing them disconnects $s$ from $t$. These are exactly the bottleneck edges.