GraphUtilities`
GraphUtilities`

# MaximalBipartiteMatching

As of Version 10, all the functionality of the GraphUtilities package is built into the Wolfram System. »

MaximalBipartiteMatching[g]

gives the maximal matching of the bipartite graph g.

# Details

• MaximalBipartiteMatching functionality is now available in the built-in Wolfram Language function FindIndependentEdgeSet.
• To use MaximalBipartiteMatching, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
• MaximalBipartiteMatching gives a maximal set of nonadjacent edges between the two vertex sets of the bipartite graph.
• The bipartite graph represented by an m×n matrix consists of the row and column vertex sets R={1,2,,m} and C={1,2,,n}, with a vertex iR and jC connected if the matrix element gij0.
• The bipartite graph represented by a rule list {i1->j1,i2->j2,} consists of vertex sets R=Union[{i1,i2,}] and C=Union[{j1,j2,}], with a vertex iR and jC connected if the rule i->j is included in the rule list.
• MaximalBipartiteMatching returns a list of index pairs {{i1,j1},,{ik,jk}}, where the number of pairs k is not larger than either vertex set.

# Examples

open allclose all

## Basic Examples(2)

A bipartite graph describing acceptable drinks for four people:

The drink each person should have, if no two people are to have the same drink:

MaximalBipartiteMatching has been superseded by FindIndependentEdgeSet:

## Applications(1)

This defines a random 30×40 sparse matrix with approximately 4% of the elements nonzero:

This finds rows and columns that are matched:

This finds unmatched rows and columns:

This orders the matrix by permuting matched rows and columns to the principal diagonal block first:

Wolfram Research (2007), MaximalBipartiteMatching, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/MaximalBipartiteMatching.html.

#### Text

Wolfram Research (2007), MaximalBipartiteMatching, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/MaximalBipartiteMatching.html.

#### BibTeX

@misc{reference.wolfram_2021_maximalbipartitematching, author="Wolfram Research", title="{MaximalBipartiteMatching}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/MaximalBipartiteMatching.html}", note=[Accessed: 25-June-2021 ]}

#### BibLaTeX

@online{reference.wolfram_2021_maximalbipartitematching, organization={Wolfram Research}, title={MaximalBipartiteMatching}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/MaximalBipartiteMatching.html}, note=[Accessed: 25-June-2021 ]}

#### CMS

Wolfram Language. 2007. "MaximalBipartiteMatching." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/GraphUtilities/ref/MaximalBipartiteMatching.html.

#### APA

Wolfram Language. (2007). MaximalBipartiteMatching. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/GraphUtilities/ref/MaximalBipartiteMatching.html