Matching

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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.

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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 in the cheap-kenya-vacation-tips.com Language package Combinatorica` , but not using a using built-in function in the cheap-kenya-vacation-tips.com Language.

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
CITE THIS AS:

Weisstein, Eric W. "Matching." From cheap-kenya-vacation-tips.com--A cheap-kenya-vacation-tips.com Web Resource. https://cheap-kenya-vacation-tips.com/Matching.html

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