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GraphUnion
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GraphUnion

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GraphUnion[g1,g2]

gives the graph union of the graphs g1 and g2.

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GraphUnion[g1,g2,…]

gives the graph union of g1, g2, ….

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GraphUnion[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

  • The graph union Graph[v1,e1]⋃Graph[v2,e2] is given by Graph[v1⋃v2,e1⋃e2].
  • GraphUnion works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Background & Context

  • GraphUnion gives a new graph obtained from a set of two or more directed or undirected graphs obtained by separately taking the union of the original vertex and edge sets. For edges with the same vertex labels in different graphs, GraphUnion keeps only one of them. The resulting graph keeps the vertex labels of the unique original edges.
  • Related functions include GraphDisjointUnion, GraphIntersection, and GraphDifference. Unlike GraphUnion, GraphDisjointUnion keeps all edges even if multiple edges exist in different graphs that have the same vertex labels. GraphIntersection gives the graph obtained from the union of vertex sets and intersection of edge sets of the original graphs. GraphDifference gives the graph obtained from the union of vertex sets of two graphs and the complement of the second graph’s edge set with respect to the first. GraphComplement gives the graph that has the same vertex set as a given graph, but with edges corresponding to absent edges in the original (and vice versa).

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Obtain the graph union of two graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-gafzdb
In[2]:=2
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https://wolfram.com/xid/0tz1rtlu-evj2e0
Out[2]=2

Highlight the graph union:

In[3]:=3
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https://wolfram.com/xid/0tz1rtlu-ea4o1m
Out[3]=3

Scope  (6)Survey of the scope of standard use cases

GraphUnion works with undirected graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-cqa97c
Out[1]=1

Directed graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-bhjscp
Out[1]=1

Multigraphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-jkqgp
Out[1]=1

Mixed graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-iy93i5
Out[1]=1

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-bndh30
Out[1]=1

GraphUnion works with more than two graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-fi43l1
Out[1]=1

Properties & Relations  (7)Properties of the function, and connections to other functions

The vertices of the graph union are the union of the vertices of the graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-e0o8hx
In[2]:=2
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https://wolfram.com/xid/0tz1rtlu-gk8fqy
Out[2]=2

The edges of the graph union are the union of the edges of the graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-czor38
In[2]:=2
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https://wolfram.com/xid/0tz1rtlu-zrwkg
In[3]:=3
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https://wolfram.com/xid/0tz1rtlu-6cs1c0
In[4]:=4
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https://wolfram.com/xid/0tz1rtlu-83gcfq
Out[4]=4

The graph union of a graph and its subgraph is isomorphic to itself:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-dan2dz
In[2]:=2
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https://wolfram.com/xid/0tz1rtlu-ko2n2b
Out[2]=2

The graph union of any simple graph and its complement is a complete graph:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-hch6zr
In[2]:=2
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https://wolfram.com/xid/0tz1rtlu-emajil
Out[2]=2

The GraphUnion of two graphs has the same vertices as GraphDifference:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-8oqxj
Out[1]=1

The GraphUnion of two graphs has the same vertices as GraphIntersection:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-byfrs
Out[1]=1

The GraphDisjointUnion can be found using GraphUnion:

In[1]:=1
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https://wolfram.com/xid/0tz1rtlu-m35yuo
In[2]:=2
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https://wolfram.com/xid/0tz1rtlu-euu882
Out[2]=2
Wolfram Research (2010), GraphUnion, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphUnion.html (updated 2015).
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Wolfram Research (2010), GraphUnion, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphUnion.html (updated 2015).

Text

Wolfram Research (2010), GraphUnion, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphUnion.html (updated 2015).

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Wolfram Research (2010), GraphUnion, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphUnion.html (updated 2015).

CMS

Wolfram Language. 2010. "GraphUnion." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/GraphUnion.html.

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Wolfram Language. 2010. "GraphUnion." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/GraphUnion.html.

APA

Wolfram Language. (2010). GraphUnion. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphUnion.html

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Wolfram Language. (2010). GraphUnion. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphUnion.html

BibTeX

@misc{reference.wolfram_2025_graphunion, author="Wolfram Research", title="{GraphUnion}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphUnion.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_graphunion, author="Wolfram Research", title="{GraphUnion}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphUnion.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_graphunion, organization={Wolfram Research}, title={GraphUnion}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphUnion.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_graphunion, organization={Wolfram Research}, title={GraphUnion}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphUnion.html}, note=[Accessed: 29-March-2025 ]}