igraph (version 0.5.2-2)

graph-isomorphism: Graph Isomorphism

Description

These functions deal with graph isomorphism.

Usage

graph.isomorphic(graph1, graph2)
graph.isomorphic.34(graph1, graph2)
graph.isomorphic.bliss(graph1, graph2, sh1="fm", sh2="fm")
graph.isomorphic.vf2(graph1, graph2)
graph.count.isomorphisms.vf2(graph1, graph2)
graph.get.isomorphisms.vf2(graph1, graph2)
graph.subisomorphic.vf2(graph1, graph2)
graph.count.subisomorphisms.vf2(graph1, graph2)
graph.get.subisomorphisms.vf2(graph1, graph2)
graph.isoclass(graph)
graph.isoclass.subgraph(graph, vids)
graph.isocreate(size, number, directed=TRUE)

Arguments

graph

A graph object.

graph1,graph2

Graph objects

size

A numeric integer giving the number of vertices in the graph to create. Only three or four are suppported right now.

number

The number of the isomorphism class of the graph to be created.

directed

Whether to create a directed graph.

sh1

Character constant, the heuristics to use in the BLISS algorithm, for graph1. See the sh argument of canonical.permutation for possible values.

sh2

Character constant, the heuristics to use in the BLISS algorithm, for graph2. See the sh argument of canonical.permutation for possible values.

vids

Numeric vector, the vertex ids of vertices to form the induced subgraph for determining the isomorphism class.

Value

• graph.isomorphic and graph.isomorphic.34 return a logical scalar, TRUE if the input graphs are isomorphic, FALSE otherwise.

  graph.isomorphic.bliss returns a named list with elements:

• iso
  A logical scalar, whether the two graphs are isomorphic.
• map12
  A numeric vector, an mapping from graph1 to graph2 if iso is TRUE, an empty numeric vector otherwise.
• map21
  A numeric vector, an mapping from graph2 to graph1 if iso is TRUE, an empty numeric vector otherwise.
• info1
  Some information about the canonical form calculation for graph1. A named list, see the return value of canonical.permutation for details.
• info2
  Some information about the canonical form calculation for graph2. A named list, see the return value of canonical.permutation for details.
• graph.isomorphic.vf2 returns a names list with three elements:
• iso
  A logical scalar, whether the two graphs are isomorphic.
• map12
  A numeric vector, an mapping from graph1 to graph2 if iso is TRUE, an empty numeric vector otherwise.
• map21
  A numeric vector, an mapping from graph2 to graph1 if iso is TRUE, an empty numeric vector otherwise.
• graph.count.isomorphisms.vf2 returns a numeric scalar, an integer, the number of isomorphic mappings between the two input graphs.

  graph.get.isomorphisms.vf2 returns a list of numeric vectors. Every numeric vector is a permutation which takes graph2 into graph1.

  graph.subisomorphic.vf2 returns a named list with three elements:

• iso
  Logical scalar, TRUE if a subgraph of graph1 is isomorphic to graph2.
• map12
  Numeric vector, empty if iso is FALSE. Otherwise a mapping from a subgraph of graph1 to graph2. -1 denotes the vertices which are not part of the mapping.
• map21
  Numeric vector, empty if iso is FALSE. Otherwise a mapping from graph2 into graph1.
• graph.count.subisomorphisms.vf2 returns a numeric scalar, an integer.

  graph.get.subisomorphisms.vf2 returns a list of numeric vectors, each numeric vector is an isomorphic mapping from graph2 to a subgraph of graph1. graph.isoclass and graph.isoclass.subgraph return a non-negative integer number.

  graph.isocreate returns a graph object.

concept

• Graph isomorphism
• Subgraph isomorphism
• VF2 algorithm
• BLISS algorithm

Details

graph.isomorphic decides whether two graphs are isomorphic. The input graphs must be both directed or both undirected. This function is a higher level interface to the other graph isomorphism decision functions. Currently it does the following:

1. If the two graphs do not agree in the number of vertices and the number of edges thenFALSEis returned.
2. Otherwise if the graphs have 3 or 4 vertices, thenigraph.isomorphic.34is called.
3. Otherwise if the graphs are directed, thenigraph.isomorphic.vf2is called.
4. Otherwiseigraph.isomorphic.blissis called.

igraph.isomorphic.34 decides whether two graphs, both of which contains only 3 or 4 vertices, are isomorphic. It works based on a precalculated and stored table.

igraph.isomorphic.bliss uses the BLISS algorithm by Junttila and Kaski, and it works for undirected graphs. For both graphs the canonical.permutation and then the permute.vertices function is called to transfer them into canonical form; finally the canonical forms are compared.

graph.isomorphic.vf2 decides whethe two graphs are isomorphic, it implements the VF2 algorithm, by Cordella, Foggia et al., see references.

graph.count.isomorphisms.vf2 counts the different isomorphic mappings between graph1 and graph2. (To count automorphisms you can supply the same graph twice, but it is better to call graph.automorphisms.) It uses the VF2 algorithm.

graph.get.isomorphisms.vf2 calculates all isomorphic mappings between graph1 and graph2. It uses the VF2 algorithm.

graph.subisomorphic.vf2 decides whether graph2 is isomorphic to some subgraph of graph1. It uses the VF2 algorithm.

graph.count.subisomorphisms.vf2 counts the different isomorphic mappings between graph2 and the subgraphs of graph1. It uses the VF2 algorithm.

graph.get.subisomorphisms.vf2 calculates all isomorphic mappings between graph2 and the subgraphs of graph1. It uses the VF2 algorithm. graph.isoclass returns the isomorphism class of a graph, a non-negative integer number. Graphs (with the same number of vertices) having the same isomorphism class are isomorphic and isomorphic graphs always have the same isomorphism class. Currently it can handle only graphs with 3 or 4 vertices.

graph.isoclass.subgraph calculates the isomorphism class of a subgraph of the input graph. Currently it only works for subgraphs with 3 or 4 vertices. graph.isocreate create a graph from the given isomorphic class. Currently it can handle only graphs with 3 or 4 vertices.

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.

See Also

graph.motifs

Examples

# create some non-isomorphic graphs
g1 <- graph.isocreate(3, 10)
g2 <- graph.isocreate(3, 11)
graph.isoclass(g1)
graph.isoclass(g2)
graph.isomorphic(g1, g2)

# create two isomorphic graphs, by
# permuting the vertices of the first 
g1 <- simplify(barabasi.game(30, m=2, directed=FALSE))
g2 <- permute.vertices(g1, sample(vcount(g1))-1)
# should be TRUE
graph.isomorphic(g1, g2)
graph.isomorphic.bliss(g1, g2)
graph.isomorphic.vf2(g1, g2)