Basic flow definitions

Residual network

Let [math]\displaystyle{ G=(V,A) }[/math] be a directed graph. Without loss of generality, we assume [math]\displaystyle{ (v,w)\in A }[/math] if, and only if, [math]\displaystyle{ (w,v)\in A }[/math]. For [math]\displaystyle{ a\in A }[/math], there are real values [math]\displaystyle{ u(a) }[/math] and [math]\displaystyle{ f(a) }[/math] such that [math]\displaystyle{ 0\leq f(a)\leq u(a) }[/math].

The residual network of [math]\displaystyle{ (G,u) }[/math] with respect to [math]\displaystyle{ f }[/math] is the pair [math]\displaystyle{ (G,u_f) }[/math] defined by [math]\displaystyle{ u_f(v,w):=u(v,w)-f(v,w)+f(w,v) }[/math] for all [math]\displaystyle{ (v,w)\in A }[/math]. The value [math]\displaystyle{ u_f(a) }[/math] is called the residual capacity of [math]\displaystyle{ a\in A }[/math] with respect to [math]\displaystyle{ f }[/math].

Therefore, the residual capacity of an arc [math]\displaystyle{ (v,w)\in A }[/math] is the amount by which the net flow from [math]\displaystyle{ v }[/math] could be changed within the capacity constraints just by changes of the flow values of [math]\displaystyle{ (v,w) }[/math] and [math]\displaystyle{ (w,v) }[/math].

Flow-augmenting path

Let [math]\displaystyle{ G=(V,A)\lt /math be a directed graph , \lt math\gt \ell(a) }[/math] and [math]\displaystyle{ u(a) }[/math] and [math]\displaystyle{ f(a)\in[\ell(a)\ldots u(a)] }[/math]