The canonical permutation brings every isomorphic graphs into the same (labeled) graph.

## Usage

```
canonical_permutation(
graph,
colors,
sh = c("fm", "f", "fs", "fl", "flm", "fsm")
)
```

## Arguments

- graph
The input graph, 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
Type of the heuristics to use for the BLISS algorithm. See details for possible values.

## Value

A list with the following members:

- labeling
The canonical permutation which takes the input graph into canonical form. A numeric vector, the first element is the new label of vertex 0, the second element for vertex 1, etc.

- info
Some information about the BLISS computation. A named list with the following members:

- "nof_nodes"
The number of nodes in the search tree.

- "nof_leaf_nodes"
The number of leaf nodes in the search tree.

- "nof_bad_nodes"
Number of bad nodes.

- "nof_canupdates"
Number of canrep updates.

- "max_level"
Maximum level.

- "group_size"
The size of the automorphism group of the input graph, as a string. The string representation is necessary because the group size can easily exceed values that are exactly representable in floating point.

## Details

`canonical_permutation()`

computes a permutation which brings the graph
into canonical form, as defined by the BLISS algorithm. All isomorphic
graphs have the same canonical form.

See the paper below for the details about BLISS. This and more information is available at http://www.tcs.hut.fi/Software/bliss/index.html.

The possible values for the `sh`

argument are:

- "f"
First non-singleton cell.

- "fl"
First largest non-singleton cell.

- "fs"
First smallest non-singleton cell.

- "fm"
First maximally non-trivially connectec non-singleton cell.

- "flm"
Largest maximally non-trivially connected non-singleton cell.

- "fsm"
Smallest maximally non-trivially connected non-singleton cell.

See the paper in references for details about these.

## 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.

## See also

`permute()`

to apply a permutation to a graph,
`graph.isomorphic()`

for deciding graph isomorphism, possibly
based on canonical labels.

Other graph isomorphism:
`count_isomorphisms()`

,
`count_subgraph_isomorphisms()`

,
`graph_from_isomorphism_class()`

,
`isomorphic()`

,
`isomorphism_class()`

,
`isomorphisms()`

,
`subgraph_isomorphic()`

,
`subgraph_isomorphisms()`

## Author

Tommi Junttila for BLISS, Gabor Csardi csardi.gabor@gmail.com for the igraph and R interfaces.

## Examples

```
## Calculate the canonical form of a random graph
g1 <- sample_gnm(10, 20)
cp1 <- canonical_permutation(g1)
cf1 <- permute(g1, cp1$labeling)
## Do the same with a random permutation of it
g2 <- permute(g1, sample(vcount(g1)))
cp2 <- canonical_permutation(g2)
cf2 <- permute(g2, cp2$labeling)
## Check that they are the same
el1 <- as_edgelist(cf1)
el2 <- as_edgelist(cf2)
el1 <- el1[order(el1[, 1], el1[, 2]), ]
el2 <- el2[order(el2[, 1], el2[, 2]), ]
all(el1 == el2)
#> [1] TRUE
```