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The article by Ford and Fulkerson (1956) with maximum flow problem established the famous theorem of maximum flow - minimum cut.

Cut: A cut defines a set of arcs whose deletion from the network causes a complete cessation of flow between origin and destination. Cutting capacity is equal to the sum of the capacities of the arcs associated. Among all possible cuts in the network, cutting with the least capacity is a minimum cut provides maximum flow on the network.

Ownership: v (f) = f (S, S ')-f (S', S).

Cut with smaller capacity called minimum cut.

compatible Any node from the source node to destination node can not exceed the capacity of any cut. Therefore, the maximum flow through the network is limited by the minimum cut capacity.

any network for maximum flow from the source node to destination node is equal to the target cutting capacity.

From this theorem the problem of finding the maximum flow in a network means finding the capacity of all cuts and choose the minimum capacity. Moreover, given the maximum flow value is not specified as this flow is distributed to ART cuts separating the source and destination node 2n-2.

Example: In a directed graph
a set of arrows S such that every directed path from s contains an arrow from S, we say that S separates as of t.

A separate court of t as a set of arrows between t as a court.

In a directed graph the minimum number of arrows between t as equal to the maximum number of disjoint directed paths of arrows that connect s to t.

NOTE: if two directed paths do not have common arrows may have common vertices. In contrast, if two paths have no common vertices have no arrows not common.

This algorithm can be used to solve models: transport of goods (supply logistics and distribution), gases and liquids flow through pipes, components or parts on assembly lines, current in electrical networks, information packets in communication networks , rail traffic, irrigation system, etc.

Ford-Fulkerson Theorem (1962): In any network, the maximum flow that flows from source to destination equals the minimum capacity cut separating the source and destination.