A matching, also called an independent edge phối, on a graph

is a set of edges of such that no two sets share a vertex in common.Bạn đang xem: Matching

It is not possible for a matching on a graph with

nodes lớn exceed edges. When a matching with edges exists, it is called a perfect matching. When a matching exists that leaves a single vertex unmatched, it is called a near-perfect matching.While not all graphs have perfect matchings, a largest matching (commonly known as a maximum matching or maximum independent edge set) exists for every graph. The kích thước of this maximum matching is called the matching number of

and is denoted .The number of matchings in a graph is sometimes called the Hosoyaindex.

A maximal independent edge mix, which is different from a maximum independent edge set, is a matching that cannot be enlarged by simply adding an edge. Such matchings are easy khổng lồ compute, but are not necessarily maximum independent edge sets. A maximal independent edge set on a general graph can be found using MaximalMatching

The blossom algorithm can be used lớn find a maximum independent edge mix in a general graph, while the simpler Hungarian maximum matching algorithm can be used for bipartite graphs. A maximum independent edge mix can be computed in the cheap-kenya-vacation-tips.com Language using FindIndependentEdgeSet

Let the number of distinct

-matchings of a graph with vertices be denoted . Then (since the empty set consisting of no edges is always a 0-matching) and , where is the edge count of .The matching polynomial is defined by

và the matching-generating polynomialby

The numbers of distinct

-matchings for various specials classes of graphs are summarized in the following table, where denotes a factorial, is a double factorial, is a binomial coefficient, and is the discrete delta function.graph | |

complete graph | |

complete bipartite graph | |

cycle graph | |

empty graph | |

path graph |

SEE ALSO: Berge"s Theorem, Blossom Algorithm, Hosoya Index, Hungarian Maximum Matching Algorithm, Independent Edge Set, Marriage Theorem, Matching-Generating Polynomial, Matching Number, Matching Polynomial, Maximal Independent Edge Set, Maximum Independent Edge Set, Near-Perfect Matching, Perfect Matching, Stable Marriage Problem

REFERENCES:

Hopcroft, J. & Karp, R. "An

Algorithm for Maximum Matching in Bipartite Graphs." SIAM J. Comput.**2**, 225-231, 1975.

Lovász, L. và Plummer, M.D. MatchingTheory. Amsterdam, Netherlands: North-Holl&, 1986.

Pemmaraju, S. and Skiemãng cầu, S. "Matching." §8.4 in Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, pp.343-351, 2003.

Xem thêm: Công An Thành Phố Hồ Chí Minh Tiếng Anh Là Gì, Từ Vựng Tiếng Anh Về Thành Phố

Skiemãng cầu, S. "Matching." §6.4 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp.240-246, 1990.

Zwiông xã, U. "Lecture Notes on: Maximum Matching in Bipartite and Non-BipartiteGraphs." 2009. https://www.cs.tau.ac.il/~zwick/grad-algo-0910/match.pdf.

Referenced on cheap-kenya-vacation-tips.com|Alpha: Matching

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