I was playing around with https://visualgo.net/en/maxflow when I realized a pattern:

enter image description here

Take this graph, for example. We notice that the min-cut divides the graph into two sets of nodes: {0, 2, 3, 6} and {1, 4, 5, 7}.

I noticed that choosing any node within {0, 2, 3, 6} as a source and {1, 4, 5, 7} as a sink, or vice versa, produces the same min-cut edges.

Is this true in general? As in, will the min-cut produced be the same for any source-sink pair from one-side of a min cut to the other?


1 Answer 1


That is an interesting observation.

However, the propostion is not true in general. For a counterexample, consider the flow network with capacities $c(\overrightarrow{AB})=2$, $c(\overrightarrow{BC})=1$ and $c(\overrightarrow{AC})=1$.

a simple flow network made with

Let $A$ be the source and $C$ be the sink. The unique min-$A$-$C$-cut is $(\{A,B\}, \{C\})$.

Let $B$ be the source and $C$ be the sink. The unique min-$B$-$C$-cut is $(\{B\}, \{A,C\})$.


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