Which min-cut does the Ford-Fulkerson algorithm find?

Question

There can be multiple min-cuts in a network. E.g:

has four min-cuts and Ford-Fulkerson finds the one "nearer" to s (the source). Can we say the same for all networks? That is, Ford-Fulkerson finds the cut nearest to the source? If true, how do we formalize the concept of "nearest to the source" in flow networks?

Answer 1

Let's represent a cut as a set of vertices that contains the source but not the sink. Minimum cuts have the property that the intersection of two minimum cuts is a minimum cut (this is true for unions also). Thus, the intersection of all minimum cuts is in a sense the minimum cut "nearest" to the source, because it is a subset of every other minimum cut.