R igraph manual pages

Minimum cut in a graph

Description

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).

Usage

min_cut(
  graph,
  source = NULL,
  target = NULL,
  capacity = NULL,
  value.only = TRUE
)

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.

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.

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.

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

Examples

g <- make_ring(100)
min_cut(g, capacity=rep(1,vcount(g)))
min_cut(g, value.only=FALSE, capacity=rep(1,vcount(g)))

g2 <- 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)

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