Compute the generating set of the automorphism group of a graph.

## Usage

``````automorphism_group(
graph,
colors = NULL,
sh = c("fm", "f", "fs", "fl", "flm", "fsm"),
details = FALSE
)``````

## Arguments

graph

The input graph, it is treated as undirected.

colors

The colors of the individual vertices of the graph; only vertices having the same color are allowed to match each other in an automorphism. When omitted, igraph uses the `color` attribute of the vertices, or, if there is no such vertex attribute, it simply assumes that all vertices have the same color. Pass NULL explicitly if the graph has a `color` vertex attribute but you do not want to use it.

sh

The splitting heuristics for the BLISS algorithm. Possible values are: ‘`f`’: first non-singleton cell, ‘`fl`’: first largest non-singleton cell, ‘`fs`’: first smallest non-singleton cell, ‘`fm`’: first maximally non-trivially connected non-singleton cell, ‘`flm`’: first largest maximally non-trivially connected non-singleton cell, ‘`fsm`’: first smallest maximally non-trivially connected non-singleton cell.

details

Specifies whether to provide additional details about the BLISS internals in the result.

## Value

When `details` is `FALSE`, a list of vertex permutations that form a generating set of the automorphism group of the input graph. When `details` is `TRUE`, a named list with two members:

generators

Returns the generators themselves

info

Additional information about the BLISS internals. See `count_automorphisms()` for more details.

## Details

An automorphism of a graph is a permutation of its vertices which brings the graph into itself. The automorphisms of a graph form a group and there exists a subset of this group (i.e. a set of permutations) such that every other permutation can be expressed as a combination of these permutations. These permutations are called the generating set of the automorphism group.

This function calculates a possible generating set of the automorphism of a graph using the BLISS algorithm. See also the BLISS homepage at http://www.tcs.hut.fi/Software/bliss/index.html. The calculated generating set is not necessarily minimal, and it may depend on the splitting heuristics used by BLISS.

## References

Tommi Junttila and Petteri Kaski: Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs, Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments and the Fourth Workshop on Analytic Algorithms and Combinatorics. 2007.

`canonical_permutation()`, `permute()`, `count_automorphisms()`

Other graph automorphism: `count_automorphisms()`

## Author

Tommi Junttila (http://users.ics.aalto.fi/tjunttil/) for BLISS, Gabor Csardi csardi.gabor@gmail.com for the igraph glue code and Tamas Nepusz ntamas@gmail.com for this manual page.

## Examples

``````
## A ring has n*2 automorphisms, and a possible generating set is one that
## "turns" the ring by one vertex to the left or right
g <- make_ring(10)
automorphism_group(g)
#> [[1]]
#> + 10/10 vertices, from dc08ff2:
#>  [1]  1 10  9  8  7  6  5  4  3  2
#>
#> [[2]]
#> + 10/10 vertices, from dc08ff2:
#>  [1]  2  3  4  5  6  7  8  9 10  1
#>
``````