# residual graph and augmenting path in max flow

I thought I understood max flow perfectly until I got to the exam and we got this. I know how to compute a maximum flow by means of the Ford-Fulkerson algorithm, specify the residual network and augmenting path. But here they gave us a graph where some of the capacities were already full (if you can tell from the picture) and I couldn't find a new augmenting path. Is there another way to solve this? A full edge, e.g. $$a \rightarrow c$$, has a residual capacity of $$0$$ in the residual network. So you can't make an augmenting path over that directed edge. However the reversed edge, $$c \rightarrow a$$ has a residual capacity of $$5$$ (since $$c_{c \rightarrow a} = 0$$ and $$f_{c \rightarrow a} = -5$$). Therefore you can create an augmenting path using the reversed edge. The residual network would be: E.g. a possible augmenting path would be $$s \rightarrow c \rightarrow a \rightarrow d \rightarrow t$$. You can increase the flow along that path by 5 and get the following flow network: This is also the maximal flow, since $$\{s, c\} - \{a, b, d, t\}$$ forms a saturated s-t-cut.