 Augmenting path

An augmenting path is a source-to-sink graph path $P$ of positive capacities in the residual network. Its bottleneck is $\Delta ={\min }_{e\in P}{c}_f(e)>0$. Add $\Delta$ to original flow on forward residual edges and subtract $\Delta$ on backward residual edges. Each internal vertex still satisfies flow conservation, and the bottleneck guarantees every capacity constraint. The resulting feasible flow has strength of a flow $|f|+\Delta$. In the unit-capacity residual network example, $\Delta =1$, and the subtraction on $a\to b$ is precisely what allows the value to rise from $1$ to $2$.