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If every directed edge capacity increases by $\epsilon$, a cut of a flow network with $m(S)$ outgoing edges gains $\epsilon m(S)$. By the max-flow min-cut theorem, the new maximum flow value isFor a finite flow network, this is a nondecreasing concave piecewise-linear function, and its active cut can change as $\epsilon$ varies. Two unit-capacity edge-disjoint two-edge source-to-sink paths gain $2\epsilon$, refuting an invariably exact $\epsilon$ increase. If there is a directed source-to-sink path, every cut has at least one outgoing edge and the increase is at least $\epsilon$; without a path it can be zero.

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.