Minimum s-t Cut

The minimum $s$-$t$ cut is the cut with the smallest capacity among all $s$-$t$ cuts. It is the cheapest way to disconnect $s$ from $t$. There might be many different cuts, but this one has the lowest cost. You are looking for the bottleneck: which edges, when removed, stop all flow with the least total capacity lost?

This is the network's vulnerability point. This cut represents the weakest point in the network between $s$ and $t$. Attack this cut, and you break the connection most efficiently.