Decreasing the capacity of some edge will decrease the max flow

Question

I am trying to find a counter example, but it doesn't seem to exist. However, could not find a proof either. Maybe someone has some idea? Here is the details:

For every s-t flow network with a non-zero max flow value, there exists an edge such that decreasing the capacity of that edge will decrease the value of the max flow. Is this true of false?

Answer 1

It's true.

Ford and Fulkerson proved the max flow min cut theorem, which basically states that the max flow of a graph is equal to the minimum cut.

Now, the minimum cut corresponds to the sum of the capacities of some set of edges in the graph. What would happen if you chose to decrease the capacity of one of those edges? (I'll let you work out the rest of the proof.)