Editing Flow network (section)

===Augmenting paths===
An '''augmenting path''' is a path {{math|(''u''<sub>1</sub>, ''u''<sub>2</sub>, ..., ''u''<sub>''k''</sub>)}} in the residual network, where {{math|''u''<sub>1</sub> {{=}} ''s''}}, {{math|''u''<sub>''k''</sub> {{=}} ''t''}}, and {{math|for all ''u''<sub>''i''</sub>, ''u''<sub>''i'' + 1</sub> (''c''<sub>''f''</sub> (''u''<sub>''i''</sub>, ''u''<sub>''i'' + 1</sub>) > 0) (1 ≤ i < k)}}. More simply, an augmenting path is an available flow path from the source to the sink. A network is at [[maximum flow]] if and only if there is no augmenting path in the residual network {{math|''G''<sub>''f''</sub>}}.

The '''bottleneck''' is the minimum residual capacity of all the edges in a given augmenting path.<ref name=":0">{{Cite book |last=Kleinberg |first=Jon |url=https://www.worldcat.org/oclc/796210667 |title=Algorithm design |date=2011 |publisher=Addison-Wesley |others=Éva Tardos |isbn=978-0-13-213108-7 |edition=2nd |location=Boston, Mass. |pages=342, 346 |oclc=796210667}}</ref> See example explained in the "Example" section of this article. The flow network is at maximum flow if and only if it has a bottleneck with a value equal to zero. If any augmenting path exists, its bottleneck weight will be greater than 0. In other words, if there is a bottleneck value greater than 0, then there is an augmenting path from the source to the sink. However, we know that if there is any augmenting path, then the network is not at maximum flow, which in turn means that, if there is a bottleneck value greater than 0, then the network is not at maximum flow.

The term "augmenting the flow" for an augmenting path means updating the flow {{mvar|f}} of each arc in this augmenting path to equal the capacity {{math|''c''}} of the bottleneck. Augmenting the flow corresponds to pushing additional flow along the augmenting path until there is no remaining available residual capacity in the bottleneck.