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I'm wondering if the only way to construct a flow network with several minimum cuts is to include at least 2 consecutive edges with the same capacity, such that any one of them will be in the minimum cut edge set.

It's clear that the simplest kind of such a network is a path with all edges of the same capacity, so in that case any one of |E| edges can be the edge separating S from T.

Is this the only way to construct such networks? If so, how can I prove it?

Pavel
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1 Answers1

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I don't think this is true:

Consider this network:

A->B capacity 4
B->C capacity 2
A->C capacity 1
C->D capacity 3

We can either have a cut (A,B,C)/(D) or (A,B)/(C,D) both with cut of 3.

enter image description here

Peter de Rivaz
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  • Oh yea, that's true, so it's all about a vertex that takes in x and pushes out x, such that the edges that go in or go out can be in a minimum cut, thanks. – Pavel Jan 15 '16 at 21:59
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    You could even have C->X having capacity 5, and X->D having capacity 3, so the min-cuts are not even adjacent. – Peter de Rivaz Jan 15 '16 at 22:00
  • that's true, my brain is still bad an coming up with network flow ideas – Pavel Jan 15 '16 at 22:02
  • but in this question they explained that min cut is unique? https://stackoverflow.com/questions/7673711/determining-the-uniqueness-of-a-min-cut – arkham knight Dec 18 '18 at 09:42
  • @user9355495 I believe in the other question they explain a method to determine whether a min cut is unique or not. – Peter de Rivaz Dec 18 '18 at 13:47
  • I do not understand the difference between these two questions properly can you kindly elaborate a little? – arkham knight Dec 18 '18 at 15:13
  • This question is asking for an example of a graph with a min-cut that is not unique. The other question is asking for an algorithm that allows a test of whether the min-cut for a graph is unique or not. – Peter de Rivaz Dec 18 '18 at 16:53