# cut

On a digraph, define a sink to be a vertex with out-degree zero and a source to be a vertex with in-degree zero. Let $G$ be a digraph with non-negative weights and with exactly one sink and exactly one source. A cut $C$ on $G$ is a subset of the edges such that every path from the source to the sink passes through an edge in $C$. In other words, if we remove every edge in $C$ from the graph, there is no longer a path from the source to the sink.

Define the weight of $C$ as

 $W_{C}=\sum_{e\in C}W(e)$

where $W(e)$ is the weight of the edge $e$.

Observe that we may achieve a trivial cut by removing all the edges of $G$. Typically, we are more interested in minimal cuts, where the weight of the cut is minimized for a particular graph.

Title cut Cut 2013-03-22 13:00:53 2013-03-22 13:00:53 vampyr (22) vampyr (22) 5 vampyr (22) Definition msc 05C20 MaximumFlowminimumCutTheorem minimum cut