Editing Flow network (section)

===Residuals===
The '''residual capacity''' of an arc {{mvar|e}} with respect to a pseudo-flow {{mvar|f}} is denoted {{math|''c''<sub>''f''</sub>}}, and it is the difference between the arc's capacity and its flow. That is, {{math|''c''<sub>''f''</sub> (''e'') {{=}} ''c''(''e'') &minus; ''f''(''e'')}}. From this we can construct a '''residual network''', denoted {{math|''G''<sub>''f''</sub> (''V'', ''E''<sub>''f''</sub>)}}, with a capacity function {{math|''c''<sub>''f''</sub>}} which models the amount of ''available'' capacity on the set of arcs in {{math|''G'' {{=}} (''V'', ''E'')}}. More specifically, capacity function {{math|''c''<sub>''f''</sub>}} of each arc {{math|(''u'', ''v'')}} in the residual network represents the amount of flow which can be transferred from {{mvar|u}} to {{mvar|v}} given the current state of the flow within the network.

This concept is used in [[Ford–Fulkerson algorithm]] which computes the [[maximum flow]] in a flow network.

Note that there can be an unsaturated path (a path with available capacity) from {{mvar|u}} to {{mvar|v}} in the residual network, even though there is no such path from {{mvar|u}} to {{mvar|v}} in the original network.{{Citation needed|date=January 2023}} Since flows in opposite directions cancel out, ''decreasing'' the flow from {{mvar|v}} to {{mvar|u}} is the same as ''increasing'' the flow from {{mvar|u}} to {{mvar|v}}.