Listing all minimum (s,t)-cuts of a directed graph, for given s and t.
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 aweight
edge attribute, then this attribute defines the edge capacities. For forcing unit edge capacities, even for graphs that have aweight
edge attribute, supplyNA
here.
Value
A list with entries:
- value
Numeric scalar, the size of the minimum cut(s).
- cuts
A list of numeric vectors containing edge ids. Each vector is a minimum (s,t)-cut.
- partition1s
A 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.
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 unweighted (=equally weighted) graphs, this is simply the number of edges.
An (s,t)-cut is minimum if it is of the smallest possible size.
References
JS Provan and DR Shier: A Paradigm for listing (s,t)-cuts in graphs, Algorithmica 15, 351--372, 1996.
See also
Other flow:
dominator_tree()
,
edge_connectivity()
,
is_min_separator()
,
is_separator()
,
max_flow()
,
min_cut()
,
min_separators()
,
min_st_separators()
,
st_cuts()
,
vertex_connectivity()
Author
Gabor Csardi csardi.gabor@gmail.com
Examples
# A difficult graph, from the Provan-Shier paper
g <- graph_from_literal(
s --+ a:b, a:b --+ t,
a --+ 1:2:3:4:5, 1:2:3:4:5 --+ b
)
st_min_cuts(g, source = "s", target = "t")
#> $value
#> [1] 2
#>
#> $cuts
#> $cuts[[1]]
#> + 2/14 edges from de7fdd1 (vertex names):
#> [1] s->a s->b
#>
#> $cuts[[2]]
#> + 2/14 edges from de7fdd1 (vertex names):
#> [1] s->b a->t
#>
#> $cuts[[3]]
#> + 2/14 edges from de7fdd1 (vertex names):
#> [1] s->a b->t
#>
#> $cuts[[4]]
#> + 2/14 edges from de7fdd1 (vertex names):
#> [1] a->t b->t
#>
#>
#> $partition1s
#> $partition1s[[1]]
#> + 1/9 vertex, named, from de7fdd1:
#> [1] s
#>
#> $partition1s[[2]]
#> + 7/9 vertices, named, from de7fdd1:
#> [1] s a 5 4 3 2 1
#>
#> $partition1s[[3]]
#> + 2/9 vertices, named, from de7fdd1:
#> [1] s b
#>
#> $partition1s[[4]]
#> + 8/9 vertices, named, from de7fdd1:
#> [1] s b a 5 4 3 2 1
#>
#>