Viapath as a maximum flow problem Let $G = (V, E)$ be a graph and $a$, $b$, $x$ $\in V \ $  different vertices. I have seen stated that the problem of finding a simple path from $a$ to $b$ passing through $x$ can be formulated as a maximum flow problem. Does anyone
have a pointer for this ? Also any references to problems that can be reformulated, in not completely obvious ways, to maximum flow problems would be valued.
The reduction is the following. 

Consider a graph as directed (that means, each edge $(u,v)$ is actually a pair of arcs pointing in the opposite directions). Split each vertex $v \in V$ to 2 different vertices $v_{in}$ and $v_{out}$ and fix the adjacent edges such that all in-arcs ends in $v_{in}$ and all out-arcs starts from $v_{out}$. Also add an arc $(v_{in}, v_{out})$ of unit capacity for each such pair. This guarantees that the flow will not pass through any original vertex more than once (which is necessary for path simplicity).

Then add a sink $t$ and arcs $(a_{out},t)$ and $(b_{out}, t)$ of unit capacity. Find a maximum flow from $x_{out}$ to $t$. If a simple path $a-x-b$ exists in the original graph, the parts $x-a$ and $x-b$ will not share a vertex (except from $x$) and the maximum flow will have size 2 so you will be able to reconstruct the answer from the flow. The opposite is true, too.

Upd.: The aforesaid was written under assumption that "simple path" means "a path with no repeated vertices". In other case there is no need for vertices splitting.