Consider a flow network and an arbitrary s, t-cut (S, T). We know that by definition s must always be on the S "side" of a cut and t is always going to be on the T "side" of the cut. Obviously, this is true for any cut. Now, consider minimum cuts. This is obviously still true for s and t, but what about other vertices in the flow network? Are there vertices that will always be on one side or the other in every minimum cut? Let's define these notions more concretely. • We say a vertex v is source-docked if v ∈ S for all minimum cuts (S, T). • We say a vertex v is sink-docked if v ∈ T for all minimum cuts (S, T). • We say a vertex v is undocked if v is neither source-docked nor sink-docked. That is, there exist minimum cuts (S, T) and (S 0 , T0 ) such that v ∈ S and v ∈ T'
Consider a flow network and an arbitrary s, t-cut (S, T). We know that by definition s must always be on the S "side" of a cut and t is always going to be on the T "side" of the cut. Obviously, this is true for any cut. Now, consider minimum cuts. This is obviously still true for s and t, but what about other vertices in the flow network? Are there vertices that will always be on one side or the other in every minimum cut?
Let's define these notions more concretely.
• We say a vertex v is source-docked if v ∈ S for all minimum cuts (S, T).
• We say a vertex v is sink-docked if v ∈ T for all minimum cuts (S, T).
• We say a vertex v is undocked if v is neither source-docked nor sink-docked. That is, there exist minimum cuts (S, T) and (S 0 , T0 ) such that v ∈ S and v ∈ T'
Give an algorithm that takes as input a flow network G and assigns each vertex to one of the three categories above.
Prove that your algorithm is correct and efficient.
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