 Vertex splitting

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.