Let $G$ be a flow network, where $c(e)$ is the capacity of an edge, and the source is $s$ and sink $t$. Define a node $v$ to be "source-like" if for every min-cut $(S,T)$ of $G$ where $S$ contains $s$, $S$ contains $v$. Find an efficient algorithm to determine the source-like nodes.
I was thinking that since Ford-Fulkerson is efficient, we can go ahead and run that to get the max flow solution, and perhaps use that to find the source-like nodes. However, as I look at examples and try to think of what to do with the flow network, I don't see how to identify source-like nodes.
Of course when you look at the residual graph at the end of Ford-Fulkerson, there is a cut where all the edges point away from $t$ and toward $s$ roughly speaking. This cut is a min-cut. But it is just one min-cut. I'm not sure how to work in the idea of all possible min-cuts. Cutting an edge seems like a bad idea. Redirecting flow in small changes seems like a bad idea.
When I think of a graph where I can trivially spot two distinct min-cuts, I think of:
s -> b weight 3 s -> c weight 3 b -> d weight 3 c -> e weight 3 d -> t weight 3 e -> t weight 3
Basically this just sets up two parallel tracks. In fact as I think about it, here I think the source-like node would be the source node.
I wonder if it's the case that in any max-flow solution, every min-cut is some set of edges which are completely used up by the max-flow?