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isisomorphic

Determine whether two graphs are isomorphic

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Syntax

tf = isisomorphic(G1,G2)
tf = isisomorphic(G1,G2,Name,Value)

Description

tf = isisomorphic(G1,G2) returns logical 1 (true) if a graph isomorphism exists between graphs G1 and G2; otherwise, it returns logical 0 (false).

example

tf = isisomorphic(G1,G2,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.

example

Examples

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Open Live Script

Create and plot two directed graphs, and then determine if they are isomorphic.

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)

Figure contains 2 axes objects. Axes object 1 contains an object of type graphplot. Axes object 2 contains an object of type graphplot.

isisomorphic(G1,G2)
ans = logical
   1
Open Live Script

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])

Figure contains an axes object. The axes object contains an object of type graphplot.

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])

Figure contains an axes object. The axes object contains an object of type graphplot.

Determine whether an isomorphism exists for G1 and G2. The result indicates that the graphs are structurally the same despite their different labels and layouts.

tf = isisomorphic(G1,G2)
tf = logical
   1
Open Live Script

Use two different comparisons to determine if there is an isomorphism relation between two graphs. One of the comparisons 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 = {'red' 'red' 'blue'}';

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

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,{'d' 'e'},'NodeColor','r')

subplot(1,2,2)
p2 = plot(G2);
highlight(p2,'c','NodeColor','r')

Figure contains 2 axes objects. Axes object 1 contains an object of type graphplot. Axes object 2 contains an object of type graphplot.

Determine if the graphs are isomorphic, ignoring the Color property. 

tf = isisomorphic(G1,G2)
tf = logical
   1

Determine if the graphs are isomorphic and preserve the value of the Color property in the comparison. In this case, there is no isomorphism since the Color property of each graph contains different numbers of 'red' and 'blue' values.

tf = isisomorphic(G1,G2,'NodeVariables','Color')
tf = logical
   0

Input Arguments

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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: tf = isisomorphic(G1,G2,'NodeVariables',{'Var1' 'Var2'})

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 comparison must preserve the specified edge variables in order to be valid. For multigraphs with multiple edges between the same two nodes, the ordering of the edge variables for the same node pair is irrelevant.

Data Types: char | string | cell

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 comparison must preserve the specified node variables in order to be valid.

Data Types: char | string | cell

More About

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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

See Also

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