Explain how to find a max $(s-t)$ flow in a network

Question

Explain how to find a max $(s-t)$ flow in a network, where some vertices are assigned capacities giving the maximum flow that can pass through those vertices.

and Illustrate this method on the following example.

Answer 1

In a general case, you can reduce the problem to the calculation of max $(s-t)$ flow in a usual flow network, by splitting each vertex $v$ with capacity $c$ into two vertices $v’$ and $v’’$ and an edge $(v’,v’’)$ of capacity $c$. Each edge $(u,v)$ ingoing to the vertex $v$ you’ll have to change by an edge $(u,v’)$ and each edge $(v,u)$ outgoing from the vertex $v$ you’ll have to change by an edge $(v’’,u)$. But in your concrete case it suffices just to change the edge capacity $5$ to $4$ to ignore the vertex capacities.