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GraphComplement[g]

gives the graph complement of the graph g.

GraphComplement[{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
History
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GraphComplement

GraphComplement[g]

gives the graph complement of the graph g.

GraphComplement[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

  • GraphComplement is also known as edge-complementary graph.
  • The graph complement has the same vertices and edges defined by two vertices being adjacent only if they are not adjacent in g.
  • GraphComplement works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples  (2)

Graph complement of cycle graphs:

Wolfram Language code: Table[CycleGraph[n], {n, 4, 6}]
Wolfram Language code: GraphComplement /@ %

Graph complement of directed graphs:

Wolfram Language code: Table[CycleGraph[n, DirectedEdges -> True, EdgeStyle -> Arrowheads[0.1]], {n, 4, 6}]
Wolfram Language code: GraphComplement /@ %

Scope  (6)

GraphComplement works with undirected graphs:

Wolfram Language code: GraphComplement[[image]]

Directed graphs:

Wolfram Language code: GraphComplement[[image]]

Multiple graphs:

Wolfram Language code: GraphComplement[[image]]

Mixed graphs:

Wolfram Language code: GraphComplement[[image]]

Use rules to specify the graph:

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

GraphComplement works with large graphs:

Wolfram Language code: g = GridGraph[{10, 10, 10, 10}];
Wolfram Language code: GraphComplement[g]//EdgeCount // Timing

Properties & Relations  (7)

The complement of a CompleteGraph is an edgeless graph:

Wolfram Language code: {CompleteGraph[5], GraphComplement[CompleteGraph[5]]}

The complement of the complement is the original graph (for simple graphs):

Wolfram Language code: NestList[GraphComplement, CycleGraph[6], 2]

The complement of the graph can be obtained from its adjacency matrix:

Wolfram Language code: g = CycleGraph[5];
Wolfram Language code: m = AdjacencyMatrix[g];
Wolfram Language code: {GraphComplement[g], AdjacencyGraph[1 - IdentityMatrix[Length[m]] - m]}
Wolfram Language code: Equal@@(AdjacencyMatrix /@ %)

An independent vertex set of the graph is a clique of its complement graph:

Wolfram Language code: g = PetersenGraph[5, 2]
Wolfram Language code: FindIndependentVertexSet[%]
Wolfram Language code: FindClique[GraphComplement[g]]

The complement of the line graph of is a Petersen graph:

Wolfram Language code: {GraphComplement[LineGraph@CompleteGraph[5]], PetersenGraph[5, 2]}
Wolfram Language code: IsomorphicGraphQ@@%

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

Wolfram Language code: g = [image];
Wolfram Language code: CompleteGraphQ[GraphUnion[g, GraphComplement[g]]]

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

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_graphcomplement, author="Wolfram Research", title="{GraphComplement}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphComplement.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_graphcomplement, organization={Wolfram Research}, title={GraphComplement}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphComplement.html}, note=[Accessed: 15-June-2026]}