title: Does a Given Network has a Unique Min-Cut?
body: <p>Let G = (V, E) be a network with s and t being the source and the sink. Let f be a maximum flow in G. Find an algorithm that determines whether there exists a unique min-cut in G.</p>
  
  <p>I have managed to find a similar question on this site:</p>
  
  <p><a href="https://stackoverflow.com/questions/7673711/determining-the-uniqueness-of-a-min-cut">Determining the uniqueness of a min-cut</a></p>
  
  <p>A summary of the answer given there:</p>
  
  <p>Find all the vertices reachable from s in the residual graph and we've found a min-cut (S,T) in G.</p>
  
  <p>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).</p>
  
  <p>This group is also a min-cut.</p>
  
  <p>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.</p>
  
  <p>I don't understand why if the cut is identical to the original cut then the cut is unique.
  Who can promise us that there is no other min-cut ?</p>
  
  <p>Thanks in advance</p>