Example - Finding Min Cut

Consider $s$ connected to $a$ (capacity $10$) and $b$ (capacity $5$). Both $a$ and $b$ connect to $t$ with capacity $8$ each.

You send $8$ through $s \to a \to t$ (limited by $a \to t$) and $5$ through $s \to b \to t$ (limited by $s \to b$). Max flow is $13$. The min cut is $\{s \to b,\, a \to t\}$ with total capacity $5 + 8 = 13$, separating $\{s, a\}$ from $\{b, t\}$.

Notice that max flow equals the min cut capacity. This confirms the max-flow min-cut theorem on a concrete example.