Question

Disclaimer: this was a homework problem. The deadline has passed now, so discussions can continue without needing to worry about that.

The problem I’m struggling with is to determine whether a particular minimum s-t cut in a graph G = (V, E) is unique. It’s simple enough to find some min-cut using a max-flow algorithm as per this example, but how would you show it’s the min-cut?

Answer 1

Ok, since you don’t want the whole answer right away, I’m gonna give you a few hints. Read as many as you feel are necessary for you, and if you give up – go ahead and read them all.

1:
The cut is unique iff there is no other min-cut.

2:
If you succeed in finding a different min-cut, then the first min-cut isn’t unique.

3:
Your link gave us one min-cut, which is all the reachable vertices from s in the residual graph. Can you think of a way to obtain a different cut, not necessarily the same?

4:
Why did we take those vertices reachable from s in particular?

5:
Maybe we can do something analogous from t?

6:
Look at the same residual graph, starting at t. Look at the group of vertices reachable from t in the reverse direction of the arrows (meaning all the vertices which can reach t).

7:
This group is also a min-cut (or actually S \ that group, to be precise).

8 (final answer):
If that cut is identical to your original cut, then there is only one. Otherwise, you just found 2 cuts, so the original one can’t possibly be unique.