# IndependentEdgeSetQ

IndependentEdgeSetQ[g,elist]

yields True if the edge list elist is an independent edge set of the graph g, and False otherwise.

# Details

• 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.
• IndependentEdgeSetQ works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

# Examples

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## Basic Examples(1)

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

## Scope(5)

Test undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

IndependentEdgeSetQ works with large graphs:

## Applications(2)

Enumerate all independent edge sets for a cycle graph:

Enumerate all subsets of edges and select the independent edge sets:

Highlight independent sets:

Enumerate all maximal independent edge sets for a wheel graph:

Find the length of a maximal independent edge set:

Enumerate all edge subsets of length 2 and select the independent edge sets:

Highlight maximal independent sets:

## Properties & Relations(3)

A largest independent edge set can be found using FindIndependentEdgeSet:

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

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

Wolfram Research (2010), IndependentEdgeSetQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IndependentEdgeSetQ.html (updated 2014).

#### Text

Wolfram Research (2010), IndependentEdgeSetQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IndependentEdgeSetQ.html (updated 2014).

#### CMS

Wolfram Language. 2010. "IndependentEdgeSetQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/IndependentEdgeSetQ.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_independentedgesetq, author="Wolfram Research", title="{IndependentEdgeSetQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/IndependentEdgeSetQ.html}", note=[Accessed: 30-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_independentedgesetq, organization={Wolfram Research}, title={IndependentEdgeSetQ}, year={2014}, url={https://reference.wolfram.com/language/ref/IndependentEdgeSetQ.html}, note=[Accessed: 30-May-2024 ]}