# stMincuts

0th

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##### List all minimum $(s,t)$-cuts of a graph

Listing all minimum $(s,t)$-cuts of a directed graph, for given $s$ and $t$.

Keywords
graphs
##### Usage
stMincuts(graph, source, target, capacity = NULL)
##### Arguments
graph
The input graph. It must be directed.
source
The id of the source vertex.
target
The id of the target vertex.
capacity
Numeric vector giving the edge capacities. If this is NULL and the graph has a weight edge attribute, then this attribute defines the edge capacities. For forcing unit edge capacities, even for graphs that have a
##### Details

Given a $G$ directed graph and two, different and non-ajacent vertices, $s$ and $t$, an $(s,t)$-cut is a set of edges, such that after removing these edges from $G$ there is no directed path from $s$ to $t$.

The size of an $(s,t)$-cut is defined as the sum of the capacities (or weights) in the cut. For unweighed (=equally weighted) graphs, this is simply the number of edges. An $(s,t)$-cut is minimum if it is of the smallest possible size.

##### Value

• A list with entries:
• valueNumeric scalar, the size of the minimum cut(s).
• cutsA list of numeric vectors containing edge ids. Each vector is a minimum $(s,t)$-cut.
• partition1sA list of numeric vectors containing vertex ids, they correspond to the edge cuts. Each vertex set is a generator of the corresponding cut, i.e. in the graph $G=(V,E)$, the vertex set $X$ and its complementer $V-X$, generates the cut that contains exactly the edges that go from $X$ to $V-X$.

##### concept

• Edge cuts
• (s,t)-cuts
• Minimum cuts
• Minimum (s,t)-cuts

##### References

JS Provan and DR Shier: A Paradigm for listing (s,t)-cuts in graphs, Algorithmica 15, 351--372, 1996.

stCuts, minimum.size.separators

• stMincuts
##### Examples
# A difficult graph, from the Provan-Shier paper
g <- graph.formula(s --+ a:b, a:b --+ t,
a --+ 1:2:3:4:5, 1:2:3:4:5 --+ b)
stMincuts(g, source="s", target="t")
Documentation reproduced from package igraph, version 0.6.5-2, License: GPL (>= 2)

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