Decide if two graphs are isomorphic

Usage

isomorphic(graph1, graph2, method = c("auto", "direct", "vf2", "bliss"), ...)

is_isomorphic_to(
  graph1,
  graph2,
  method = c("auto", "direct", "vf2", "bliss"),
  ...
)

Arguments

graph1: The first graph.
graph2: The second graph.
method: The method to use. Possible values: ‘auto’, ‘direct’, ‘vf2’, ‘bliss’. See their details below.
...: Additional arguments, passed to the various methods.

Value

Logical scalar, TRUE if the graphs are isomorphic.

‘auto’ method

It tries to select the appropriate method based on the two graphs. This is the algorithm it uses:

If the two graphs do not agree on their order and size (i.e. number of vertices and edges), then return FALSE.
If the graphs have three or four vertices, then the ‘direct’ method is used.
If the graphs are directed, then the ‘vf2’ method is used.
Otherwise the ‘bliss’ method is used.

‘direct’ method

This method only works on graphs with three or four vertices, and it is based on a pre-calculated and stored table. It does not have any extra arguments.

‘vf2’ method

This method uses the VF2 algorithm by Cordella, Foggia et al., see references below. It supports vertex and edge colors and have the following extra arguments:

vertex.color1, vertex.color2: Optional integer vectors giving the colors of the vertices for colored graph isomorphism. If they are not given, but the graph has a “color” vertex attribute, then it will be used. If you want to ignore these attributes, then supply NULL for both of these arguments. See also examples below.
edge.color1, edge.color2: Optional integer vectors giving the colors of the edges for edge-colored (sub)graph isomorphism. If they are not given, but the graph has a “color” edge attribute, then it will be used. If you want to ignore these attributes, then supply NULL for both of these arguments.

‘bliss’ method

Uses the BLISS algorithm by Junttila and Kaski, and it works for undirected graphs. For both graphs the canonical_permutation() and then the permute() function is called to transfer them into canonical form; finally the canonical forms are compared. Extra arguments:

sh: Character constant, the heuristics to use in the BLISS algorithm for graph1 and graph2. See the sh argument of canonical_permutation() for possible values.

sh defaults to ‘fm’.

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.

LP Cordella, P Foggia, C Sansone, and M Vento: An improved algorithm for matching large graphs, Proc. of the 3rd IAPR TC-15 Workshop on Graphbased Representations in Pattern Recognition, 149–159, 2001.

Examples

# create some non-isomorphic graphs
g1 <- graph_from_isomorphism_class(3, 10)
g2 <- graph_from_isomorphism_class(3, 11)
isomorphic(g1, g2)
#> [1] FALSE

# create two isomorphic graphs, by permuting the vertices of the first
g1 <- sample_pa(30, m = 2, directed = FALSE)
g2 <- permute(g1, sample(vcount(g1)))
# should be TRUE
isomorphic(g1, g2)
#> [1] TRUE
isomorphic(g1, g2, method = "bliss")
#> [1] TRUE
isomorphic(g1, g2, method = "vf2")
#> [1] TRUE

# colored graph isomorphism
g1 <- make_ring(10)
g2 <- make_ring(10)
isomorphic(g1, g2)
#> [1] TRUE

V(g1)$color <- rep(1:2, length = vcount(g1))
V(g2)$color <- rep(2:1, length = vcount(g2))
# consider colors by default
count_isomorphisms(g1, g2)
#> [1] 10
# ignore colors
count_isomorphisms(g1, g2,
  vertex.color1 = NULL,
  vertex.color2 = NULL
)
#> [1] 20