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In the mathematical discipline of graph theory a matching or edge independent set in a graph is a set of edges without common vertices. It may also be an entire graph consisting of edges without common vertices.

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Wikipedia about matching

In the mathematical discipline of graph theory a matching or edge independent set in a graph is a set of edges without common vertices. It may also be an entire graph consisting of edges without common vertices.

Definition

Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex.

We say that a vertex is matched if it is incident to an edge in the matching. Otherwise the vertex is unmatched.

A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it is no longer a matching, that is, M is maximal if it is not a proper subset of any other matching in graph G. In other words, a matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M.

A maximum matching is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number of a graph is the size of a maximum matching. Note that every maximum matching must be maximal, but not every maximal matching must be maximum.

A perfect matching is a matching which covers all vertices of the graph. That is, every vertex of the graph is incident to exactly one edge of the matching. Every perfect matching is maximum and hence maximal. In some literature, the term complete matching is used.

Given a matching M,

  • an alternating path is a path in which the edges belong alternatively to the matching and not to the matching.

.

  • an augmenting path is an alternating path that starts from and ends on free (unmatched) vertices.

One can prove that a matching is maximum if and only if it does not have any augmenting path. (This result is sometimes called Berge's Lemma).

Matchings in bipartite graphs

Matching problems are often concerned with bipartite graphs. Finding a maximum bipartite matching , chapter 3. (often called a maximum cardinality bipartite matching) in a bipartite graph G=(V=(X,Y),E) is perhaps the simplest problem. The augmenting path algorithm finds it by finding an augmenting path from each x \in X to Y and adding it to the matching if it exists. As each path can be found in O(E) time, the running time is O(V E). This solution is equivalent to adding a super source s with edges to all vertices in X, and a super sink t with edges from all vertices in Y, and finding a maximal flow from s to t. All edges with flow from X to Y then constitute a maximum matching. An improvement over this is the Hopcroft-Karp algorithm, which runs in O(\sqrt{V} E) time.

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Rummy - Wikipedia, the free encyclopedia

Rummy. From Wikipedia, the free encyclopedia (Redirected from Matching card game) ... card games notable for gameplay based on the matching of similar playing cards. ...en.wikipedia.org/wiki/Matching_card_game

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News about Matching

Proliferation champ wants matching deal

The Economic TimesPakistan Prime Minister Yousaf Raza Gilani said Thursday his country is keen to have a deal similar to the India-US nuclear accord.http://economictimes.indiatimes.com/News/PoliticsNation/Prol...

SlimBlade Media Notebook Set

PC WorldThe Kensington SlimBlade Media Notebook Set is inarguably a stylish group of wireless input devices. Its three-piece matching... http://www.pcworld.com/article/151791/.html?tk=rss_news

Budget cuts into facade grants

The Indianapolis StarNOBLESVILLE -- The city has paid half the cost to upgrade nearly a dozen historical buildings in downtown Noblesville this year through a matching grant program, but the initiative could be scaled back next year.http://www.indystar.com/apps/pbcs.dll/article?AID=/20081002/...

Hopefuls await women's play-off fate

UEFA NewsThe ten teams through to the UEFA WOMEN'S EURO 2009™ play-offs are awaiting Monday's draw at 12.00CET in Nyon but one two-legged tie is already decided, matching Spain and the Netherlands.http://www.uefa.com/competitions/woco/news/kind=1/newsid=757...

Cowboys on 5-game skid without interception

KSLA-TV ShreveportIRVING, Texas (AP) - Counting last season's finale the Dallas Cowboys have gone five straight games without an interception by their defense, matching the longest drought in team history.http://www.ksla.com/Global/story.asp?S=9116307

News about Matching

More women boldly go, solo or matched with companion

Atlanta Journal ConstitutionThe Internet-based matching service uses a series of 120 questions to help pair solo women travelers together for upcoming adventures. Spence, who was widowed herself 11 years ago, said single women are much more adventuresome than they were three ... http://www.ajc.com/homes/content/travel/otherdestinations/us...

Major breakthrough for Weibring

Quincy Herald-WhigThe victory is Weibring's first on the Champions Tour since the 2007 3M Championship. It's his fifth Champions Tour victory, matching his total number of wins he had on the PGA Tour. For more on this story, pick up Monday's edition of The Herald-Whighttp://www.whig.com/story/sports/Weibring-Wins-10-12

New Test Shows Computers Not Yet Able To Fool Humans, But Getting ...

All Headline NewsThe machine closest to matching Turing's record of 30 percent was a computer named Elbot, who was able to trick 20 percent of the human interrogators into thinking they were communicating with a person. Despite falling just short of Turing's record ... http://www.allheadlinenews.com/articles/7012626258

Vandy tries to rebound off 1st loss of season

Rocky Mountain NewsNASHVILLE, Tenn. (AP) _ Well, the Vanderbilt Commodores have been here before. Now their challenge is simple: Prove they really are different from Vandy teams of the past 25 years. The Commodores blew a chance at matching the best start in 80 years ... http://www.rockymountainnews.com/news/2008/oct/12/vandy-trie...

Editorial: State should back Duluth rail project

Minneapolis Star TribuneHe has his eye on a proposed Minneapolis-to-Duluth high-speed line as a potential beneficiary of the bill's $1.9 billion in 80/20 matching grants for new state passenger rail service. In that, Oberstar is not alone. For years, eight Minnesota city ... http://www.startribune.com/opinion/editorials/30810914.html?...

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Matching Sox - Turning Red Sox Singles into Doubles

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Cross-matching - Wikipedia, the free encyclopedia

Cross-matching, in transfusion medicine, refers to the testing that is performed to determine the compatibility of a donated unit of blood with its intended recipient.http://en.wikipedia.org/wiki/Cross-matching

NAMI | Matching Gifts

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Answers for Matching

Jimmy L

What is the minimum matching in a bipartite graph?

Jimmy Lhttp://en.wikipedia.org/wiki/Matching Please give me best answer thanks!
xboxplaya001

What are the benefits of matching your skins ph?

xboxplaya001matching the skins acidity will give you a less chance of getting blemishes... some people with high acidity have a high risk of getting white spots on their skin therefore they can't use regular soap...
Mrs. Cleaver's Beaver

Do you ever feel like throwing all your socks out instead of matching them?

Mrs. Cleaver's BeaverYes, I sure do, I remedied this problem by buying all white socks. Sure makes it a lot easier because they all match.=)
C S

How can I find a matching rim for my truck?

C S
Gladicouldhelpu

Is a Safe Harbor 401k matching contribution to a shareholder of an S-Corp a K-1 add back item?

GladicouldhelpuI've searched Revenue Ruling 91-26 and Publication 15B and can find no mention of a matching 401k contribution listed as a fringe benefit. The matching contribution is a deduction for the S Corpora...

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