# exists (u,v) edge with positive capacity and there is not path from $s$ to $u$. and $(u,v)$ is with full capacity in some maximal flow

Given a network flow and there exists (u,v) edge with positive capacity and there is not path from $$s$$ to $$u$$. and $$(u,v)$$ is with full capacity in some maximal flow.

I've had this questions with true or false and I thought it was false but apparently it is true. Can someone please explain me how this is possible?

$$V = \{s,t,u,v\}$$ and $$E = \{(s,t), (u,s), (u,v), (v,u)\}$$ with each edge of capacity 1.
You can send a flow of 1 via $$(u,v)$$, $$(v,u)$$, and $$(s,t)$$. It follows all properties of a valid flow. It does not have a path from $$s$$ to $$u$$. Also, the flow via $$(u,v)$$ is at its full capacity.