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FindIndependentEdgeSet[g]

finds an independent edge set of the graph g with a maximum number of edges.

FindIndependentEdgeSet[{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  
See Also
Related Guides
History
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FindIndependentEdgeSet

FindIndependentEdgeSet[g]

finds an independent edge set of the graph g with a maximum number of edges.

FindIndependentEdgeSet[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

  • An independent edge set is also known as a matching.
  • An independent edge set is a set of edges that are never incident to the same vertex.
  • FindIndependentEdgeSet returns a list of edges.
  • FindIndependentEdgeSet works with undirected graphs, directed graphs, weighted graphs, and multigraphs.

Examples

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Basic Examples  (1)

Find an independent edge set in a graph:

Wolfram Language code: g = [image];
Wolfram Language code: FindIndependentEdgeSet[%]

Show the edge set:

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

Scope  (6)

FindIndependentEdgeSet works with undirected graphs:

Wolfram Language code: FindIndependentEdgeSet[[image]]

Directed graphs:

Wolfram Language code: FindIndependentEdgeSet[[image]]

Weighted graphs:

Wolfram Language code: FindIndependentEdgeSet[[image]]

Multigraphs:

Wolfram Language code: FindIndependentEdgeSet[[image]]

Use rules to specify the graph:

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

FindIndependentEdgeSet works with large graphs:

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

Applications  (3)

A company has a number of different jobs. Each employee is suited for some of these jobs, and each person can perform at most one job at a time:

Wolfram Language code: edges = Table[Subscript[P, i]Subscript[J, j], {i, 1, 6}, {j, Union[RandomInteger[{1, 5}, 2]]}]//Flatten
Wolfram Language code: coords = Join[Table[Subscript[P, i] -> {0, i - 3}, {i, 6}], Table[Subscript[J, j] -> {3, j - 2.5}, {j, 5}]];
Wolfram Language code: Graph[edges, VertexShapeFunction -> "Name", VertexCoordinates -> coords]

Maximize the number of jobs that can be performed simultaneously:

Wolfram Language code: FindIndependentEdgeSet[%]

Given a set of women, each of whom has a preference for some subset of men, find a maximal matching where only matches that agree with preferences are allowed:

Wolfram Language code: edges = Table[Subscript[W, i]Subscript[M, j], {i, 1, 4}, {j, Union[RandomInteger[{1, 4}, 2]]}]//Flatten
Wolfram Language code: coords = Join[Table[Subscript[W, i] -> {0, i - 1}, {i, 4}], Table[Subscript[M, j] -> {2, j - 1}, {j, 4}]];
Wolfram Language code: Graph[edges, VertexShapeFunction -> "Name", VertexCoordinates -> coords]

Compute a maximal matching:

Wolfram Language code: FindIndependentEdgeSet[%]

An art history department would like to offer six courses. There are eight professors, each of whom is willing to teach certain courses. Find a maximal matching where professors only teach courses they are interested in teaching:

Wolfram Language code: edges = Table[Subscript[C, i]Subscript[P, j], {i, 1, 7}, {j, Union[RandomInteger[{1, 8}, 3]]}]//Flatten;
Wolfram Language code: coor = Join[Table[Subscript[C, i] -> {0, i - 0.5}, {i, 7}], Table[Subscript[P, j] -> {4, j - 1}, {j, 8}]];
Wolfram Language code: Graph[edges, VertexShapeFunction -> "Name", VertexCoordinates -> coor]

Match preferences to courses:

Wolfram Language code: FindIndependentEdgeSet[%]

Properties & Relations  (3)

Test whether a set of edges is an independent edge set using IndependentEdgeSetQ:

Wolfram Language code: PetersenGraph[5, 2]
Wolfram Language code: IndependentEdgeSetQ[%, FindIndependentEdgeSet[%]]

Bipartite graphs have independent edge sets and vertex covers of equal length:

Wolfram Language code: g = CompleteGraph[{2, 3}]
Wolfram Language code: Length[FindVertexCover[g]] == Length[FindIndependentEdgeSet[g]]

For a graph without isolated vertices, the sum of the size of the independent edge set and the size of the edge cover is equal to the number of vertices:

Wolfram Language code: g = PetersenGraph[5, 2]
Wolfram Language code: Length[FindEdgeCover[g]] + Length[FindIndependentEdgeSet[g]] == VertexCount[g]
Wolfram Research (2010), FindIndependentEdgeSet, Wolfram Language function, https://reference.wolfram.com/language/ref/FindIndependentEdgeSet.html (updated 2015).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_findindependentedgeset, author="Wolfram Research", title="{FindIndependentEdgeSet}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FindIndependentEdgeSet.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_findindependentedgeset, organization={Wolfram Research}, title={FindIndependentEdgeSet}, year={2015}, url={https://reference.wolfram.com/language/ref/FindIndependentEdgeSet.html}, note=[Accessed: 13-June-2026]}