Let $G = (V,E)$ be a directed graph with source $s$ and sink $t$ and $s \neq t$. For each edge $e \in E$, we have $c(e) \in \Bbb N$.
also, we are given a max flow function $f$ on that network.
Let $R_f$ be the residual network that represents the max-flow $f$.
I would like to prove that the subset formed by all the nodes reachable from $s$ in $R_f$ is included in every subset $S'$ from any min-cut $(S',T')$.
so, if we will notate the group of nodes that are reachable from $s$ as $S$, I need to prove that $S \subseteq S'$.
I tried to write a Proof by contradiction but I'm just getting stuck every time trying to analyze the Min-Cut after removing a certain node.
thanks!
Let $\lambda$ be the capacity of a minimum $st-$cut. Let $v$ be a vertex in $S$. Add an arc from $v$ to $t$ with capacity $1$. Then, one can augment the flow $f$ to a flow $f'$ whose flow value is $\lambda + 1$ (by taking an augmenting path going from $s$ to $v$ and using the new arc to $t$). Then consider any cut $(X,V \setminus X)$ with $s \in X$, $v, t \notin X$. The existence of $f'$ implies that $(X, V \setminus X)$ has capacity at least $\lambda+1$. Hence for any $st$-cut $(S',T')$ in the original graph with capacity $\lambda$, $v \in S'$.