Type the residual-capacity definitions for max flow

Answer

forward residual: c_f(u, v) = c(u, v) - f(u, v) backward residual: c_f(v, u) = f(u, v) edge in G_f iff: c_f(u, v) > 0 augment by: c_f(p) = min over edges on p

Forward residual is the unused capacity; backward residual lets you cancel existing flow. Pushing the bottleneck c_f(p) along an augmenting path raises |f| by that amount.