Maximum flow with vertex capacities

The maximum flow problem with internal vertex capacities reduces to the usual edge-capacitated problem by splitting each constrained vertex into input and output copies. The connecting edge carries its entire through-flow. Feasible flows correspond under this construction, preserving the source-to-sink value; the Ford-Fulkerson algorithm and max-flow min-cut theorem therefore apply to the transformed network.

Replace a vertex $v$ by $v_{\mathrm{in}},v_{\mathrm{out}}$ and an arc between them carrying its vertex capacity. Incoming arcs enter $v_{\mathrm{in}}$ and outgoing arcs leave $v_{\mathrm{out}}$. A path must cross that internal arc to use the vertex. This converts node-capacitated maximum flow to an ordinary flow network. For undirected mixed-failure cuts, normalize the source side so $v_{\mathrm{out}}$ inside implies $v_{\mathrm{in}}$ inside; then each original link contributes at most one crossing orientation.