Consider a flow network G = (V,E) with positive edge capacities {c(e)}. Let f: E Ro...

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Consider a flow network G = (V,E) with positive edge capacities {c(e)}. Let f: E Ro be a maximum flow in G, and G be the residual graph. Denote by S the set of nodes reachable from s in G and by T the set of nodes from which t is reachable in G. That is, S = {u there is a directed path from s to u in G}, == : {v there is a directed path from v to t in G}. : Prove that V = SUT if and only if G has a unique s-t minimum cut (an s-t cut whose capacity is strictly less than the capacity of any other s-t cut). Consider a flow network G = (V,E) with positive edge capacities {c(e)}. Let f: E Ro be a maximum flow in G, and G be the residual graph. Denote by S the set of nodes reachable from s in G and by T the set of nodes from which t is reachable in G. That is, S = {u there is a directed path from s to u in G}, == : {v there is a directed path from v to t in G}. : Prove that V = SUT if and only if G has a unique s-t minimum cut (an s-t cut whose capacity is strictly less than the capacity of any other s-t cut).