Cut (graph theory)

In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.

In a flow network, an s–t cut is a cut that requires the source and the sink to be in different subsets, and its cut-set only consists of edges going from the source's side to the sink's side. The capacity of an s–t cut is defined as the sum of capacity of each edge in the cut-set.

A cut ${\displaystyle C=(S,T)}$ is a partition of ${\displaystyle V}$ of a graph ${\displaystyle G=(V,E)}$ into two subsets S and T. The cut-set of a cut ${\displaystyle C=(S,T)}$ is the set ${\displaystyle \{(u,v)\in E|u\in S,v\in T\}}$ of edges that have one endpoint in S and the other endpoint in T. If s and t are specified vertices of the graph G, then an s–t cut is a cut in which s belongs to the set S and t belongs to the set T.

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