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Subgraph[g,{v1,v2,…}]

gives the subgraph of the graph g generated by the vertices vi.

Subgraph[g,{e1,e2,…}]

gives the subgraph generated by the edges ej.

Subgraph[g,patt]

gives the subgraph generated by the vertices and edges that match the pattern patt.

Subgraph[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Neat Examples  
See Also
Related Guides
History
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Subgraph

Subgraph[g,{v1,v2,…}]

gives the subgraph of the graph g generated by the vertices vi.

Subgraph[g,{e1,e2,…}]

gives the subgraph generated by the edges ej.

Subgraph[g,patt]

gives the subgraph generated by the vertices and edges that match the pattern patt.

Subgraph[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

  • The subgraph generated by the vertices v1,v2,… includes the vertices vi and all edges connecting them in the original graph g.
  • The subgraph generated by the edges e1,e2,… includes the edges ej and all edges connecting vertices vi of ej in the original graph g.
  • Subgraph works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples  (1)

Give a subgraph of a complete graph:

Wolfram Language code: g = CompleteGraph[5]
Wolfram Language code: h = Subgraph[%, {1, 2, 3}]

Highlight the subgraph within the main graph:

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

Scope  (7)

Subgraph works with undirected graphs:

Wolfram Language code: Subgraph[[image], {1, 3, 5, 6}]

Directed graphs:

Wolfram Language code: Subgraph[[image], {1, 3, 6}]

Multigraph:

Wolfram Language code: Subgraph[[image], {1, 3, 5, 6}]

Mixed graph:

Wolfram Language code: Subgraph[[image], {1, 3, 5, 6}]

Use rules to specify the graph:

Wolfram Language code: Subgraph[{1 -> 3, 2 -> 1, 3 -> 6, 4 -> 6, 1 -> 5, 5 -> 4, 6 -> 1}, {1, 3, 6}]

Use patterns to select subgraphs:

Wolfram Language code: g = PetersenGraph[5, 2]

Find the subgraph with edges that are incident to 3:

Wolfram Language code: HighlightGraph[g, Subgraph[g, _3]]

Find the subgraph with vertices less than 6:

Wolfram Language code: HighlightGraph[g, Subgraph[g, _ ? ((# > 5)&)]]

Subgraph works with large graphs:

Wolfram Language code: g = GridGraph[{10, 10, 10, 10}];
Wolfram Language code: Subgraph[g, _ ? EvenQ]//VertexCount//Timing

Applications  (2)

Select random subgraphs of size 5:

Wolfram Language code: RandomSubgraph[g_ ? GraphQ, s_] := Subgraph[g, RandomSample[VertexList[g], s]]
Wolfram Language code: g = [image];
Wolfram Language code: Table[HighlightGraph[g, RandomSubgraph[g, 5]], {4}]

Find all the subgraphs of the graph g that are isomorphic to the graph h:

Wolfram Language code: {g, h} = {PetersenGraph[5, 2], CycleGraph[5]}

Enumerate all subsets of vertices that have the same number of vertices as h:

Wolfram Language code: s = Subsets[Range[VertexCount[g]], {VertexCount[h]}];

Select the subgraphs that are isomorphic to h:

Wolfram Language code: Select[Subgraph[g, #]& /@ s, IsomorphicGraphQ[#, h]&];

Highlight them:

Wolfram Language code: HighlightGraph[g, #]& /@ %

Properties & Relations  (3)

The vertices of subgraphs are subsets of the original vertices:

Wolfram Language code: {g = PetersenGraph[5, 2], h = Subgraph[g, {1, 3, 4}]}
Wolfram Language code: MemberQ[VertexList[g], #]& /@ VertexList[h]

The edges of subgraphs are subsets of the original edges:

Wolfram Language code: {g = PetersenGraph[5, 2], h = Subgraph[g, {1, 3, 4}]}
Wolfram Language code: MemberQ[EdgeList[g], #]& /@ EdgeList[h]

The subgraph of a complete graph is a complete graph:

Wolfram Language code: g = CompleteGraph[8]
Wolfram Language code: HighlightGraph[g, h = Subgraph[g, Range[5]]]
Wolfram Language code: CompleteGraphQ[h]

Neat Examples  (1)

Subtract random neighborhoods from a grid:

Wolfram Language code: g = GridGraph[{25, 50}];
Wolfram Language code: Subgraph[g, Complement[VertexList[g], VertexComponent[g, RandomSample[VertexList[g], 25], 1]], AnnotationRules -> Inherited]
Wolfram Research (2010), Subgraph, Wolfram Language function, https://reference.wolfram.com/language/ref/Subgraph.html (updated 2015).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_subgraph, author="Wolfram Research", title="{Subgraph}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/Subgraph.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_subgraph, organization={Wolfram Research}, title={Subgraph}, year={2015}, url={https://reference.wolfram.com/language/ref/Subgraph.html}, note=[Accessed: 12-June-2026]}