I came across a question with notation like this:
$G$ does not contain any subgraph of the form $v^→_← w$.
I think this means that there is no way to push flow from v to w. Given this is a bipartite graph (because network flow), aren't these nodes useless? This leads me to believe I am misinterpreting the notation.
It means there is no "two-cycle": a pair of arcs in opposite directions between the same pair of nodes.
Regarding "bipartite graph (because network flow)": network flow problems are also defined in graphs that aren't bipartite.