`min_cut()`

calculates the minimum st-cut between two vertices in a graph
(if the `source`

and `target`

arguments are given) or the minimum
cut of the graph (if both `source`

and `target`

are `NULL`

).

## Arguments

- graph
The input graph.

- source
The id of the source vertex.

- target
The id of the target vertex (sometimes also called sink).

- capacity
Vector giving the capacity of the edges. If this is

`NULL`

(the default) then the`capacity`

edge attribute is used.- value.only
Logical scalar, if

`TRUE`

only the minimum cut value is returned, if`FALSE`

the edges in the cut and a the two (or more) partitions are also returned.

## Value

For `min_cut()`

a nuieric constant, the value of the minimum
cut, except if `value.only = FALSE`

. In this case a named list with
components:

- value
Numeric scalar, the cut value.

- cut
Numeric vector, the edges in the cut.

- partition1
The vertices in the first partition after the cut edges are removed. Note that these vertices might be actually in different components (after the cut edges are removed), as the graph may fall apart into more than two components.

- partition2
The vertices in the second partition after the cut edges are removed. Note that these vertices might be actually in different components (after the cut edges are removed), as the graph may fall apart into more than two components.

## Details

The minimum st-cut between `source`

and `target`

is the minimum
total weight of edges needed to remove to eliminate all paths from
`source`

to `target`

.

The minimum cut of a graph is the minimum total weight of the edges needed
to remove to separate the graph into (at least) two components. (Which is to
make the graph *not* strongly connected in the directed case.)

The maximum flow between two vertices in a graph is the same as the minimum
st-cut, so `max_flow()`

and `min_cut()`

essentially calculate the same
quantity, the only difference is that `min_cut()`

can be invoked without
giving the `source`

and `target`

arguments and then minimum of all
possible minimum cuts is calculated.

For undirected graphs the Stoer-Wagner algorithm (see reference below) is used to calculate the minimum cut.

## References

M. Stoer and F. Wagner: A simple min-cut algorithm,
*Journal of the ACM*, 44 585-591, 1997.

## See also

`max_flow()`

for the related maximum flow
problem, `distances()`

, `edge_connectivity()`

,
`vertex_connectivity()`

Other flow:
`dominator_tree()`

,
`edge_connectivity()`

,
`is_min_separator()`

,
`is_separator()`

,
`max_flow()`

,
`min_separators()`

,
`min_st_separators()`

,
`st_cuts()`

,
`st_min_cuts()`

,
`vertex_connectivity()`

## Examples

```
g <- make_ring(100)
min_cut(g, capacity = rep(1, vcount(g)))
#> [1] 2
min_cut(g, value.only = FALSE, capacity = rep(1, vcount(g)))
#> $value
#> [1] 2
#>
#> $cut
#> + 2/100 edges from f2aedf8:
#> [1] 1--2 2--3
#>
#> $partition1
#> + 1/100 vertex, from f2aedf8:
#> [1] 2
#>
#> $partition2
#> + 99/100 vertices, from f2aedf8:
#> [1] 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
#> [20] 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
#> [39] 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
#> [58] 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77
#> [77] 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
#> [96] 97 98 99 100
#>
g2 <- make_graph(c(1, 2, 2, 3, 3, 4, 1, 6, 6, 5, 5, 4, 4, 1))
E(g2)$capacity <- c(3, 1, 2, 10, 1, 3, 2)
min_cut(g2, value.only = FALSE)
#> $value
#> [1] 1
#>
#> $cut
#> + 1/7 edge from 654bf9f:
#> [1] 2->3
#>
#> $partition1
#> + 1/6 vertex, from 654bf9f:
#> [1] 2
#>
#> $partition2
#> + 5/6 vertices, from 654bf9f:
#> [1] 1 3 4 5 6
#>
```