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GraphIntersection[g1,g2]

gives the graph intersection of the graphs g1 and g2.

GraphIntersection[g1,g2,…]

gives the graph intersection of g1, g2, … .

GraphIntersection[{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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GraphIntersection

GraphIntersection[g1,g2]

gives the graph intersection of the graphs g1 and g2.

GraphIntersection[g1,g2,…]

gives the graph intersection of g1, g2, … .

GraphIntersection[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

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

Examples

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Basic Examples  (1)

Obtain the graph intersection of two graphs:

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

Highlight the graph intersection:

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

Scope  (5)

GraphIntersection works with undirected graphs:

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

Directed graphs:

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

Multigraphs:

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

Mixed graphs:

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

Use rules to specify the graph:

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

Properties & Relations  (8)

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

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

The edges of the graph intersection are the intersection 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[GraphIntersection[g1, g2]] == Intersection[EdgeList[g1], EdgeList[g2], SameTest -> edgeSameQ]

The graph intersection of any graph and itself is isomorphic to the original graph:

Wolfram Language code: g = [image];
Wolfram Language code: IsomorphicGraphQ[g, GraphIntersection[g, g]]

The graph intersection of a graph and its subgraph has the same edges as its subgraph:

Wolfram Language code: g = [image];
Wolfram Language code: h = Subgraph[g, Range[5]]
Wolfram Language code: EdgeList[GraphIntersection[g, h]] == EdgeList[h]

The intersection of any graph and its complement is an empty graph:

Wolfram Language code: g = [image];
Wolfram Language code: EmptyGraphQ[GraphIntersection[g, GraphComplement[g]]]

The intersection of two graphs with different names is an empty graph:

Wolfram Language code: EmptyGraphQ[GraphIntersection[[image], [image]]]

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

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

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

Wolfram Language code: VertexList[GraphIntersection[[image], [image]]] == VertexList[GraphDifference[[image], [image]]]
Wolfram Research (2010), GraphIntersection, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphIntersection.html (updated 2015).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_graphintersection, author="Wolfram Research", title="{GraphIntersection}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphIntersection.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_graphintersection, organization={Wolfram Research}, title={GraphIntersection}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphIntersection.html}, note=[Accessed: 13-June-2026]}