verifying minimum capacity of cut

Question

So I have a cut $(P,P')$ on some network and its capacity is $13$. Now I'm told to assume that the current flow on the network is the max flow, is the cut of minimum capacity? So far all we've learned is that the max flow is at most equal to the capacity of the cut but I we haven't dealt with minimum capacity of cut. Thank you in advance.

Answer 1

If you have a flow in the network of 13 units then the cut is saturated and 13 will be equal to the maximum flow. Any other cut must then have an equal or larger capacity. So any cut where the flow is equal to the capacity is a minimal cut.