isomorphism

Compute isomorphism between two graphs

Syntax

P = isomorphism(G1,G2)

P = isomorphism(___,Name,Value)

[P,edgeperm] = isomorphism(___)

Description

P = isomorphism(G1,G2) computes a graph isomorphism equivalence relation between graphs G1 and G2, if one exists. If no isomorphism exists, then P is an empty array.

P = isomorphism(___,Name,Value) specifies additional options with one or more name-value pair arguments. For example, you can specify 'NodeVariables' and a list of node variables to indicate that the isomorphism must preserve these variables to be valid.

[P,edgeperm] = isomorphism(___) additionally returns a vector of edge permutations, edgeperm. This output enables you to preserve edge variables when working with multigraphs.

Examples

Compute Isomorphism Permutation

Create and plot two directed graphs, and then calculate the isomorphism relation between them.

G1 = digraph([1 1 1 2 3 4],[2 3 4 4 4 1]);
G2 = digraph([3 3 3 2 1 4],[1 4 2 3 2 2]);
subplot(1,2,1)
plot(G1)
subplot(1,2,2)
plot(G2)

p = isomorphism(G1,G2)

p = 4×1

     3
     1
     4
     2

The result indicates that reordernodes(G2,p) has the same structure as G1.

Compute Isomorphism Between Graphs with Different Labels and Layouts

Create and plot two graphs, G1 and G2.

G1 = graph([1 1 1 2 2 3 3 4 5 5 7 7],[2 4 5 3 6 4 7 8 6 8 6 8]);
plot(G1,'XData',[1 4 4 1 2 3 3 2],'YData',[4 4 1 1 3 3 2 2])

G2 = graph({'a' 'a' 'a' 'b' 'b' 'b' 'c' 'c' 'c' 'd' 'd' 'd'}, ...
    {'g' 'h' 'i' 'g' 'h' 'j' 'g' 'i' 'j' 'h' 'i' 'j'});
plot(G2,'XData',[1 2 2 2 1 2 1 1],'YData',[4 4 3 2 3 1 2 1])

Compute the isomorphism relation between the graphs, if one exists. The result indicates that the graph nodes can be permuted to represent the same graph despite their different labels and layouts.

p = isomorphism(G1,G2)

p = 8×1

     1
     2
     5
     3
     4
     7
     6
     8

Compute Isomorphism and Preserve Node Properties

Compute two different isomorphism relations between two graphs. One of the relations preserves a node property, while the other ignores it.

Create two similar graphs. Add a node property Color to each of the graphs.

G1 = graph({'d' 'e' 'f'},{'e' 'f' 'd'});
G1.Nodes.Color = {'blue' 'red' 'red'}';

G2 = graph({'a' 'b' 'c'},{'b' 'c' 'a'});
G2.Nodes.Color = {'red' 'red' 'blue'}';

Plot the graphs side-by-side in the same figure. Color the nodes red that have Color = 'red'.

subplot(1,2,1)
p1 = plot(G1);
highlight(p1,{'e' 'f'},'NodeColor','r')

subplot(1,2,2)
p2 = plot(G2);
highlight(p2,{'a' 'b'},'NodeColor','r')

Compute the isomorphism between the graphs, ignoring the Color property.

p = isomorphism(G1,G2)

p = 3×1

     1
     2
     3

Compute the isomorphism again, but this time preserve the value of the Color property in the comparison. isomorphism returns a different permutation that preserves the Color property.

p = isomorphism(G1,G2,'NodeVariables','Color')

p = 3×1

     3
     1
     2

View the nodes in G1 and G2 that the isomorphism matches together.

[G1.Nodes.Name, G2.Nodes.Name(p)]

ans = 3x2 cell
    {'d'}    {'c'}
    {'e'}    {'a'}
    {'f'}    {'b'}

Input Arguments

G1,G2 — Input graphs (as separate arguments)

graph objects | digraph objects

Input graphs, specified as separate arguments of graph or digraph objects. Use graph to create an undirected graph or digraph to create a directed graph.

G1 and G2 must be both graph objects or both digraph objects.

Example: G1 = graph(1,2)

Example: G1 = digraph([1 2],[2 3])

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: P = isomorphism(G1,G2,'NodeVariables',{'Var1' 'Var2'})

EdgeVariables — Edge variables to preserve

character vector | string scalar | cell array of character vectors | string array

Edge variables to preserve, specified as the comma-separated pair consisting of 'EdgeVariables' and a character vector, string scalar, cell array of character vectors, or string array. Use this option to specify one or more edge variables that are in both G1.Edges and G2.Edges. The isomorphism must preserve the specified edge variables in order to be valid.

If G is a multigraph, then you can specify the second output edgeperms to enable reordering edge variables.

Data Types: char | string | cell

NodeVariables — Node variables to preserve

character vector | string scalar | cell array of character vectors | string array

Node variables to preserve, specified as the comma-separated pair consisting of 'NodeVariables' and a character vector, string scalar, cell array of character vectors, or string array. Use this option to specify one or more node variables that are in both G1.Nodes and G2.Nodes. The isomorphism must preserve the specified node variables in order to be valid.

Data Types: char | string | cell

Output Arguments

P — Permutation vector for isomorphism

column vector | []

Permutation vector for isomorphism, returned as a column vector when an isomorphism exists or as the empty array [] when an isomorphism does not exist. If P is not empty, then reordernodes(G2,P) has the same structure as G1.

edgeperm — Edge permutation

column vector

Edge permutation, returned as a column vector. When working with multigraphs, the edge permutation vector enables you to preserve edge variables specified by the 'EdgeVariables' name-value pair. Use these commands to reorder the edge variables of repeated edges:

[p,edgeperm] = isomorphism(g1,g2,'EdgeVariables',edgevars);
g2perm = reordernodes(g2, p);
g2perm.Edges(:, 2:end) = g2perm.Edges(edgeperm, 2:end);

More About

Graph Isomorphism

Two graphs, G1 and G2, are isomorphic if there exists a permutation of the nodes P such that reordernodes(G2,P) has the same structure as G1.

Two graphs that are isomorphic have similar structure. For example, if a graph contains one cycle, then all graphs isomorphic to that graph also contain one cycle.

Extended Capabilities

Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

Version History

Introduced in R2016b