Suppose there is a directed graph $G=(V,E)$, with source $s$ and sink $t$, and I compute the max flow on it. Then I know that I can find a min-cut $(A,B)$, by letting $A$ be the set of vertices that are reachable from the source $s$.
My question is is this set $A$ the smallest possible $s$-component? I think it is. To be precise, does there exist a subset of vertices $A^*$, such that $(A^*, V\setminus A^*)$ is a min-cut, and for all other possible subsets of vertices $A$ such that $(A, V\setminus A)$ is a min-cut, we have $size(A^*) \leq size(A)$.
Furthermore, for any other min-cut $(C,D)$, do we have $A \subset C$?. I also think this is the case. But I can't prove it.
Any intuition / hints are much appreciated!
Definition of $s$-component: The min-cut is the partition of the vertices of $G$ into two disjoint sets $(A,B)$, where $A$ contains the source $s$ and $B$ contains the sink $t$. The $s$-component w.r.t. a min-cut is then the set $A$.
By "smallest" $s$-component I mean $s$-component with smallest cardinality. I'm wondering if there can be several different minimal $s$-component w.r.t. set inclusion, i.e. $s$-components with same cardinality, but are not equal as sets. Equivalently, wondering if there is a minimum $s$-component; a set of vertices that is in ALL $s$-components.