This is not a home work question.

Is that correct to say in a flow network, if min cut is unique then max flow must also be unique?

Assume we have two max flow in a network. Then for each max flow, we can get a min cut and they are surely different.

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    – Raphael
    Oct 20, 2016 at 20:45

1 Answer 1


I do not agree with the downvoters of your question. This questions seems like a legit misunderstanding of the statement of the min-cost-max-flow theorem.

The theorem states that the value of the sum of edge weights in a minimum $s$-$t$ cut of a graph is equal to the value of the maximum $s$-$t$ flow in the graph interpreted as a flow network. It does not establish a one-to-one correspondence between flows and cuts.

In fact, this correspondence cannot be established in general. Here is a counter example of an unweighted undirected graph (all weights/capacities are unit):


$1$-$5$ max-flow/min-cut is equal to one. While there is a single minimum cut ({1 - 2}). There are a lot of possible maximum flows (take the unit of flow entering 2 and distribute it between 3 and 4 as you like).


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