Vertex Capacities

Some problems impose capacity limits on nodes, not edges. For example, a node can handle at most $c$ units of flow. Convert vertex capacity to edge capacity: split each node $v$ into two nodes $v_{\text{in}}$ and $v_{\text{out}}$.

Add an edge $(v_{\text{in}}, v_{\text{out}})$ with capacity $c$. All incoming edges go to $v_{\text{in}}$, all outgoing edges leave from $v_{\text{out}}$. This enforces the vertex capacity constraint.