Finding the Min-Cut

After computing max flow, the residual graph reveals the min-cut. Run a BFS or DFS from the source on the residual graph. Let $S$ be the set of reachable nodes. Let $T$ be the remaining nodes. The edges from $S$ to $T$ form the min-cut.

These edges are saturated (flow equals capacity), so they have zero residual capacity. This gives you both the max flow value and the actual edges that form the bottleneck.