The vertex connectivity of a graph or two vertices, this is recently also called group cohesion.

## Usage

```
vertex_connectivity(graph, source = NULL, target = NULL, checks = TRUE)
vertex_disjoint_paths(graph, source = NULL, target = NULL)
# S3 method for igraph
cohesion(x, checks = TRUE, ...)
```

## Arguments

- graph, x
The input graph.

- source
The id of the source vertex, for

`vertex_connectivity()`

it can be`NULL`

, see details below.- target
The id of the target vertex, for

`vertex_connectivity()`

it can be`NULL`

, see details below.- checks
Logical constant. Whether to check that the graph is connected and also the degree of the vertices. If the graph is not (strongly) connected then the connectivity is obviously zero. Otherwise if the minimum degree is one then the vertex connectivity is also one. It is a good idea to perform these checks, as they can be done quickly compared to the connectivity calculation itself. They were suggested by Peter McMahan, thanks Peter.

- ...
Ignored.

## Details

The vertex connectivity of two vertices (`source`

and `target`

) in
a graph is the minimum number of vertices that must be deleted to
eliminate all (directed) paths from `source`

to `target`

.
`vertex_connectivity()`

calculates this quantity if both the
`source`

and `target`

arguments are given and they're not
`NULL`

.

The vertex connectivity of a pair is the same as the number of different (i.e. node-independent) paths from source to target, assuming no direct edges between them.

The vertex connectivity of a graph is the minimum vertex connectivity of all
(ordered) pairs of vertices in the graph. In other words this is the minimum
number of vertices needed to remove to make the graph not strongly
connected. (If the graph is not strongly connected then this is zero.)
`vertex_connectivity()`

calculates this quantity if neither the
`source`

nor `target`

arguments are given. (I.e. they are both
`NULL`

.)

A set of vertex disjoint directed paths from `source`

to `vertex`

is a set of directed paths between them whose vertices do not contain common
vertices (apart from `source`

and `target`

). The maximum number of
vertex disjoint paths between two vertices is the same as their vertex
connectivity in most cases (if the two vertices are not connected by an
edge).

The cohesion of a graph (as defined by White and Harary, see references), is
the vertex connectivity of the graph. This is calculated by
`cohesion()`

.

These three functions essentially calculate the same measure(s), more
precisely `vertex_connectivity()`

is the most general, the other two are
included only for the ease of using more descriptive function names.

## References

White, Douglas R and Frank Harary 2001. The Cohesiveness of
Blocks In Social Networks: Node Connectivity and Conditional Density.
*Sociological Methodology* 31 (1) : 305-359.

## See also

Other flow:
`dominator_tree()`

,
`edge_connectivity()`

,
`is_min_separator()`

,
`is_separator()`

,
`max_flow()`

,
`min_cut()`

,
`min_separators()`

,
`min_st_separators()`

,
`st_cuts()`

,
`st_min_cuts()`

## Author

Gabor Csardi csardi.gabor@gmail.com

## Examples

```
g <- sample_pa(100, m = 1)
g <- delete_edges(g, E(g)[100 %--% 1])
g2 <- sample_pa(100, m = 5)
g2 <- delete_edges(g2, E(g2)[100 %--% 1])
vertex_connectivity(g, 100, 1)
#> [1] 1
vertex_connectivity(g2, 100, 1)
#> [1] 5
vertex_disjoint_paths(g2, 100, 1)
#> [1] 5
g <- sample_gnp(50, 5 / 50)
g <- as.directed(g)
g <- induced_subgraph(g, subcomponent(g, 1))
cohesion(g)
#> [1] 2
```