# Will the Ford-Fulkerson Algorithm always return the same min-cut for any source-sink from one side of the min-cut to the other?

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

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 $$c(\overrightarrow{AC})=1$$. 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 $$(\{B\}, \{A,C\})$$.