Editing Max-flow min-cut theorem (section)

===Generalized max-flow min-cut theorem===
In addition to edge capacity, consider there is capacity at each vertex, that is, a mapping <math>c:V\to\R^+</math> denoted by {{math|''c''(''v'')}}, such that the flow {{math|&thinsp;''f''&thinsp;}} has to satisfy not only the capacity constraint and the conservation of flows, but also the vertex capacity constraint

:<math>\forall v \in V \setminus \{s,t\} : \qquad \sum\nolimits_{\{u\in V\mid (u,v)\in E\}} f_{uv} \le c(v).</math>

In other words, the amount of ''flow'' passing through a vertex cannot exceed its capacity. Define an ''s-t cut'' to be the set of vertices and edges such that for any path from ''s'' to ''t'', the path contains a member of the cut. In this case, the ''capacity of the cut'' is the sum of the capacity of each edge and vertex in it.

In this new definition, the '''generalized max-flow min-cut theorem''' states that the maximum value of an s-t flow is equal to the minimum capacity of an s-t cut in the new sense.