title: Decreasing the capacity of some edge will decrease the max flow
body: <p>I am trying to find a counter example, but it doesn't seem to exist. However, could not find a proof either. Maybe someone has some idea? Here is the details:</p>
  
  <p>For every s-t flow network with a non-zero max flow value, there exists an edge such that decreasing the capacity of that edge will decrease the value of the max flow. Is this true of false?</p>