Characterizations of max-balanced flows (Q1201811)

Let \(G=(V,A)\) be a directed graph. Then a flow for \(G\) is an arbitrary real-valued function defined on the arc set \(A\). A flow \(f\) is called max-balanced if, for any \(\varnothing\subsetneqq W\subsetneqq V\), it holds \(\max_{x\in W^ +}f(x)=\max_{x\in W^ -}f(x)\), where \(W^ +\) \((W^ -)\) is the set of all arcs \(x=(u,v)\) with \(v\in W^ +\), \(u\notin W^ +\) \((u\in W^ -,v\notin W^ -)\). Ten characterizations of max- balanced flows are given.