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

Question

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?

Answer 1

Since you just have to satisfy the conservation of flow property and capacity property. You can consider the following example:

$V = \{s,t,u,v\}$ 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.