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List all (s,t)-cuts in a directed graph.

Usage

st_cuts(graph, source, target)

Arguments

graph

The input graph. It must be directed.

source

The source vertex.

target

The target vertex.

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 VX, generates the cut that contains exactly the edges that go from X to VX.

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.

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 61e8ab3 (vertex names):
#> [1] a->b
#> 
#> $cuts[[2]]
#> + 1/4 edge from 61e8ab3 (vertex names):
#> [1] b->c
#> 
#> $cuts[[3]]
#> + 1/4 edge from 61e8ab3 (vertex names):
#> [1] c->d
#> 
#> $cuts[[4]]
#> + 1/4 edge from 61e8ab3 (vertex names):
#> [1] d->e
#> 
#> 
#> $partition1s
#> $partition1s[[1]]
#> + 1/5 vertex, named, from 61e8ab3:
#> [1] a
#> 
#> $partition1s[[2]]
#> + 2/5 vertices, named, from 61e8ab3:
#> [1] a b
#> 
#> $partition1s[[3]]
#> + 3/5 vertices, named, from 61e8ab3:
#> [1] a b c
#> 
#> $partition1s[[4]]
#> + 4/5 vertices, named, from 61e8ab3:
#> [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 476c1c8 (vertex names):
#> [1] s->a s->b
#> 
#> $cuts[[2]]
#> + 2/10 edges from 476c1c8 (vertex names):
#> [1] s->a b->t
#> 
#> $cuts[[3]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t a->1 a->2 a->3
#> 
#> $cuts[[4]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t a->1 a->2 3->b
#> 
#> $cuts[[5]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t a->1 a->3 2->b
#> 
#> $cuts[[6]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t a->1 2->b 3->b
#> 
#> $cuts[[7]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t a->2 a->3 1->b
#> 
#> $cuts[[8]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t a->2 1->b 3->b
#> 
#> $cuts[[9]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t a->3 1->b 2->b
#> 
#> $cuts[[10]]
#> + 5/10 edges from 476c1c8 (vertex names):
#> [1] s->b a->t 1->b 2->b 3->b
#> 
#> $cuts[[11]]
#> + 2/10 edges from 476c1c8 (vertex names):
#> [1] a->t b->t
#> 
#> 
#> $partition1s
#> $partition1s[[1]]
#> + 1/7 vertex, named, from 476c1c8:
#> [1] s
#> 
#> $partition1s[[2]]
#> + 2/7 vertices, named, from 476c1c8:
#> [1] s b
#> 
#> $partition1s[[3]]
#> + 2/7 vertices, named, from 476c1c8:
#> [1] s a
#> 
#> $partition1s[[4]]
#> + 3/7 vertices, named, from 476c1c8:
#> [1] s a 3
#> 
#> $partition1s[[5]]
#> + 3/7 vertices, named, from 476c1c8:
#> [1] s a 2
#> 
#> $partition1s[[6]]
#> + 4/7 vertices, named, from 476c1c8:
#> [1] s a 2 3
#> 
#> $partition1s[[7]]
#> + 3/7 vertices, named, from 476c1c8:
#> [1] s a 1
#> 
#> $partition1s[[8]]
#> + 4/7 vertices, named, from 476c1c8:
#> [1] s a 1 3
#> 
#> $partition1s[[9]]
#> + 4/7 vertices, named, from 476c1c8:
#> [1] s a 1 2
#> 
#> $partition1s[[10]]
#> + 5/7 vertices, named, from 476c1c8:
#> [1] s a 1 2 3
#> 
#> $partition1s[[11]]
#> + 6/7 vertices, named, from 476c1c8:
#> [1] s a 1 2 3 b
#> 
#>