# Min-cut in a network with zero flow from source to sink

The max-flow min-cut theorem guarantees that the min-cut of a directed network equals the maximum flow. And we can compute $$S$$ and $$T$$, are disjoint subsets containing source node and sink node respectively from the residual graph.

What will be $$S$$ and $$T$$ if the max-flow from source to sink is 0, that is, there is no directed path from $$s$$ to $$t$$? Is $$S$$ going to be the singleton set {s} or its the empty set $$\phi$$ and $$T$$ be the vertex set $$V$$?

• Have you checked the relevant definitions? Have you constructed a few simple examples? – Apass.Jack Mar 13 at 2:55
• @Apass.Jack, my gut feeling is, if the flow from source to sink is zero, then $S$ should be the singleton $S$ and $T$ should be the other vertices of the graph, but I can't seem to find anything in the literature to support this. All discussions assume there is some non-zero flow from source to sink. I want to use the min-cut as a subroutine in another algorithm and this result will greatly affect the outcome of the algorithm. – Ozymandais Mar 13 at 3:12

Consider the following flow network with source $$s$$ and sink $$t$$, where the capacity of every edge is 1. The max-flow from source to sink is 0. The s-t cut $$(\{s,A\}, \{B,t\})$$ is a minimum cut since the only connecting edge $$(B, A)$$ goes from sink side to source side.
$$s \longrightarrow A \longleftarrow B \longrightarrow t$$