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GraphDifference

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gives the graph difference of the graphs g1 and g2.

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GraphDifference[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

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

Examples

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

Obtain the graph difference of two graphs:

In[31]:=31
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https://wolfram.com/xid/0i1qh1frm-gafzdb
In[32]:=32
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https://wolfram.com/xid/0i1qh1frm-evj2e0
Out[32]=32

Highlight the graph difference:

In[33]:=33
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https://wolfram.com/xid/0i1qh1frm-ea4o1m
Out[33]=33

Scope  (5)Survey of the scope of standard use cases

GraphDifference works with undirected graphs:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-cc12xh
Out[1]=1

Directed graphs:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-kbwby
Out[1]=1

Multigraphs:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-jkqgp
Out[1]=1

Mixed graphs:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-dbshxt
Out[1]=1

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-bndh30
Out[1]=1

Properties & Relations  (6)Properties of the function, and connections to other functions

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

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-e0o8hx
In[2]:=2
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https://wolfram.com/xid/0i1qh1frm-gk8fqy
Out[2]=2

The edges of the graph difference are the complement of the edges of the graphs:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-djgbmo
In[2]:=2
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https://wolfram.com/xid/0i1qh1frm-zrwkg
In[3]:=3
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https://wolfram.com/xid/0i1qh1frm-eu2sds
Out[3]=3

The graph difference of any graph and itself is an empty graph:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-cqvir3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0i1qh1frm-elkhmh
Out[2]=2

The graph difference of any graph and its CompleteGraph is isomorphic to the complement of the graph:

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-b1xu9g
In[2]:=2
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https://wolfram.com/xid/0i1qh1frm-0v8ne
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-jrp2tv
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0i1qh1frm-eo7hj5
Out[1]=1
Wolfram Research (2010), GraphDifference, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphDifference.html (updated 2015).
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Wolfram Research (2010), GraphDifference, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphDifference.html (updated 2015).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_graphdifference, author="Wolfram Research", title="{GraphDifference}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphDifference.html}", note=[Accessed: 08-July-2025 ]}

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@misc{reference.wolfram_2025_graphdifference, author="Wolfram Research", title="{GraphDifference}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphDifference.html}", note=[Accessed: 08-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_graphdifference, organization={Wolfram Research}, title={GraphDifference}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphDifference.html}, note=[Accessed: 08-July-2025 ]}

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@online{reference.wolfram_2025_graphdifference, organization={Wolfram Research}, title={GraphDifference}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphDifference.html}, note=[Accessed: 08-July-2025 ]}