Residual Network

Given flow network G = (V, E) and flow f, the residual network of G induced by f is where

$$E_f \;=\; \{(u,v) \;\in\; V x V \;\mid\; c_f(u,v) \;>\; 0\}$$

(u,v) $\in E_f$ is a residual edge; i.e., any edge that can admit more flow.

If there is a path from s to t in the residual network, then it is an augmenting path and indicates where flow can increase.