Residual Network (flow)

Let $(G, c, s, t)$ be a flow network and $f$ be a flow. Further assume $G$ has no dual edges. Define the $G^f = (V,E^f)$ where

$$E^f = \{(v,w) \in E \vert f(v,w) < c(v,w)\} \cup \{(w,v) \vert (v,w) \in E, f(v,w) > 0\}$$

and capacity

$$c^f(v,w) = \begin{cases} c(v,w) - f(v,w) & \mbox{if } (v,w) \in E \mbox{ and } f(v,w) < c(v,w)\\ f(w,v) & \mbox{if } (w,v) \in E \mbox{ and } f(w,v) > 0 \end{cases}.$$

The let the residual flow network be $(G^f, c^f, s, t)$.