List all (s,t)-cuts in a directed graph.
Value
A list with entries:
- cuts
A list of numeric vectors containing edge ids. Each vector is an \((s,t)\)-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 \(t\).
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_min_cuts(),
vertex_connectivity()
Author
Gabor Csardi csardi.gabor@gmail.com
Examples
# A very simple graph
g <- graph_from_literal(a -+ b -+ c -+ d -+ e)
st_cuts(g, source = "a", target = "e")
#> $cuts
#> $cuts[[1]]
#> + 1/4 edge from 5f4150b (vertex names):
#> [1] a->b
#>
#> $cuts[[2]]
#> + 1/4 edge from 5f4150b (vertex names):
#> [1] b->c
#>
#> $cuts[[3]]
#> + 1/4 edge from 5f4150b (vertex names):
#> [1] c->d
#>
#> $cuts[[4]]
#> + 1/4 edge from 5f4150b (vertex names):
#> [1] d->e
#>
#>
#> $partition1s
#> $partition1s[[1]]
#> + 1/5 vertex, named, from 5f4150b:
#> [1] a
#>
#> $partition1s[[2]]
#> + 2/5 vertices, named, from 5f4150b:
#> [1] a b
#>
#> $partition1s[[3]]
#> + 3/5 vertices, named, from 5f4150b:
#> [1] a b c
#>
#> $partition1s[[4]]
#> + 4/5 vertices, named, from 5f4150b:
#> [1] a b c d
#>
#>
# A somewhat more difficult graph
g2 <- graph_from_literal(
s --+ a:b, a:b --+ t,
a --+ 1:2:3, 1:2:3 --+ b
)
st_cuts(g2, source = "s", target = "t")
#> $cuts
#> $cuts[[1]]
#> + 2/10 edges from 3153f8d (vertex names):
#> [1] s->a s->b
#>
#> $cuts[[2]]
#> + 2/10 edges from 3153f8d (vertex names):
#> [1] s->a b->t
#>
#> $cuts[[3]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t a->1 a->2 a->3
#>
#> $cuts[[4]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t a->1 a->2 3->b
#>
#> $cuts[[5]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t a->1 a->3 2->b
#>
#> $cuts[[6]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t a->1 2->b 3->b
#>
#> $cuts[[7]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t a->2 a->3 1->b
#>
#> $cuts[[8]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t a->2 1->b 3->b
#>
#> $cuts[[9]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t a->3 1->b 2->b
#>
#> $cuts[[10]]
#> + 5/10 edges from 3153f8d (vertex names):
#> [1] s->b a->t 1->b 2->b 3->b
#>
#> $cuts[[11]]
#> + 2/10 edges from 3153f8d (vertex names):
#> [1] a->t b->t
#>
#>
#> $partition1s
#> $partition1s[[1]]
#> + 1/7 vertex, named, from 3153f8d:
#> [1] s
#>
#> $partition1s[[2]]
#> + 2/7 vertices, named, from 3153f8d:
#> [1] s b
#>
#> $partition1s[[3]]
#> + 2/7 vertices, named, from 3153f8d:
#> [1] s a
#>
#> $partition1s[[4]]
#> + 3/7 vertices, named, from 3153f8d:
#> [1] s a 3
#>
#> $partition1s[[5]]
#> + 3/7 vertices, named, from 3153f8d:
#> [1] s a 2
#>
#> $partition1s[[6]]
#> + 4/7 vertices, named, from 3153f8d:
#> [1] s a 2 3
#>
#> $partition1s[[7]]
#> + 3/7 vertices, named, from 3153f8d:
#> [1] s a 1
#>
#> $partition1s[[8]]
#> + 4/7 vertices, named, from 3153f8d:
#> [1] s a 1 3
#>
#> $partition1s[[9]]
#> + 4/7 vertices, named, from 3153f8d:
#> [1] s a 1 2
#>
#> $partition1s[[10]]
#> + 5/7 vertices, named, from 3153f8d:
#> [1] s a 1 2 3
#>
#> $partition1s[[11]]
#> + 6/7 vertices, named, from 3153f8d:
#> [1] s a 1 2 3 b
#>
#>