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

## Usage

```
edge_connectivity(graph, source = NULL, target = NULL, checks = TRUE)
edge_disjoint_paths(graph, source, target)
adhesion(graph, checks = TRUE)
```

## Arguments

- graph
The input graph.

- source
The id of the source vertex, for

`edge_connectivity()`

it can be`NULL`

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

`edge_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 edge 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.

`edge_connectivity()`

Edge connectivity

The edge connectivity of a pair of vertices (`source`

and
`target`

) is the minimum number of edges needed to remove to eliminate
all (directed) paths from `source`

to `target`

.
`edge_connectivity()`

calculates this quantity if both the `source`

and `target`

arguments are given (and not `NULL`

).

The edge connectivity of a graph is the minimum of the edge connectivity of
every (ordered) pair of vertices in the graph. `edge_connectivity()`

calculates this quantity if neither the `source`

nor the `target`

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

).

`edge_disjoint_paths()`

The maximum number of edge-disjoint paths between two vertices

A set of paths between two vertices is called edge-disjoint if they do not share any edges. The maximum number of edge-disjoint paths are calculated by this function using maximum flow techniques. Directed paths are considered in directed graphs.

A set of edge disjoint paths between two vertices is a set of paths between them containing no common edges. The maximum number of edge disjoint paths between two vertices is the same as their edge connectivity.

When there are no direct edges between the source and the target, the number of vertex-disjoint paths is the same as the vertex connectivity of the two vertices. When some edges are present, each one of them contributes one extra path.

`adhesion()`

Adhesion of a graph

The adhesion of a graph is the minimum number of edges needed to remove to obtain a graph which is not strongly connected. This is the same as the edge connectivity of the graph.

## All three functions

The three functions documented on this page calculate similar properties,
more precisely the most general is `edge_connectivity()`

, the others are
included only for having more descriptive function names.

## References

Douglas R. White and Frank Harary: The cohesiveness of blocks in social networks: node connectivity and conditional density, TODO: citation

## See also

Other flow:
`dominator_tree()`

,
`is_min_separator()`

,
`is_separator()`

,
`max_flow()`

,
`min_cut()`

,
`min_separators()`

,
`min_st_separators()`

,
`st_cuts()`

,
`st_min_cuts()`

,
`vertex_connectivity()`

## Author

Gabor Csardi csardi.gabor@gmail.com

## Examples

```
g <- sample_pa(100, m = 1)
g2 <- sample_pa(100, m = 5)
edge_connectivity(g, 100, 1)
#> [1] 1
edge_connectivity(g2, 100, 1)
#> [1] 5
edge_disjoint_paths(g2, 100, 1)
#> [1] 5
g <- sample_gnp(50, 5 / 50)
g <- as.directed(g)
g <- induced_subgraph(g, subcomponent(g, 1))
adhesion(g)
#> [1] 1
```