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GraphDifference[g1,g2]

gives the graph difference of the graphs g1 and g2.

GraphDifference[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Related Guides
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GraphDifference

GraphDifference[g1,g2]

gives the graph difference of the graphs g1 and g2.

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)

Obtain the graph difference of two graphs:

Wolfram Language code: g = [image];h = [image];
Wolfram Language code: x = GraphDifference[g, h]

Highlight the graph difference:

Wolfram Language code: HighlightGraph[g, %, GraphHighlightStyle -> "Thick"]

Scope  (5)

GraphDifference works with undirected graphs:

Wolfram Language code: GraphDifference[[image], [image]]

Directed graphs:

Wolfram Language code: GraphDifference[[image], [image]]

Multigraphs:

Wolfram Language code: GraphDifference[[image], [image]]

Mixed graphs:

Wolfram Language code: GraphDifference[[image], [image]]

Use rules to specify the graph:

Wolfram Language code: GraphDifference[{1 -> 2, 2 -> 3, 3 -> 1, 4 -> 3, 2 -> 4}, {1 -> 2, 3 -> 2, 4 -> 3, 4 -> 1}]

Properties & Relations  (6)

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

Wolfram Language code: g1 = [image];g2 = [image];
Wolfram Language code: VertexList[GraphDifference[g1, g2]] == Union[VertexList[g1], VertexList[g2]]

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

Wolfram Language code: g1 = [image];g2 = [image];
Wolfram Language code: edgeSameQ[e1_, e2_] := SameQ[e1, e2] || SameQ[e1, Reverse[e2]]
Wolfram Language code: EdgeList[GraphDifference[g1, g2]] == Complement[EdgeList[g1], EdgeList[g2], SameTest -> edgeSameQ]

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

Wolfram Language code: GraphDifference[[image], [image]]
Wolfram Language code: EmptyGraphQ[%]

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

Wolfram Language code: g = [image];
Wolfram Language code: IsomorphicGraphQ[GraphDifference[CompleteGraph[6], g], GraphComplement[g]]

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

Wolfram Language code: VertexList[GraphDifference[[image], [image]]] == VertexList[GraphUnion[[image], [image]]]

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

Wolfram Language code: VertexList[GraphDifference[[image], [image]]] == VertexList[GraphIntersection[[image], [image]]]
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).

CMS

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

BibTeX

@misc{reference.wolfram_2026_graphdifference, author="Wolfram Research", title="{GraphDifference}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphDifference.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_graphdifference, organization={Wolfram Research}, title={GraphDifference}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphDifference.html}, note=[Accessed: 13-June-2026]}