Concepts of bipartite graphs

This will be a long path. This path starts from a bipartite graph and its end point will be a quadratic matching problem (QAP ).

Recently I have been studying the assignment problem. I have seen binary graphs and searched for a lot of information about binary graphs on the Internet. However, I found that the definition of the concept of binary graphs in Chinese documents is not very accurate. Now I want to sort out what I need to know as follows.

(1) Binary Graph Definition

In the mathematical field of graph theory,Bipartite Graph(OrBigraph) Is a graph whose vertices can be divided into twoDisjointSets and such that everyEdgeConnects a vertex in to one in; that is, and are each independent sets. equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. [1]

One often writes to denote a bipartite graph whose partition has the parts and. [1]

(2) Matching

The concept of matching is not exclusive to a bipartite graph. It is a concept in graph theory.

Given a graphG= (V,E),Matching MInGIs a set of pairwiseNon-adjacentEdges; that is, no two edges share a common vertex. [2]

From this definition, we can see that a match is a subset of the G edge of the graph.

A vertex isMatched(OrSaturated) If it is an endpoint of one of the edges in the matching. Otherwise the vertex isUnmatched. [2]

(3) maximal matching

A matching m is said to be maximal if M is not properly contained in any other matching. [4]

This means that when we can no longer add any edge to m, it is not a match, and m is maximal.

It is easy to find a maximal matching.

(4) maximum matching

AMaximum matchingIs a matching that contains the largest possible number of edges. There may be using Maximum Matchings. [2]

(5) Augmenting Path (augmented path) [4]

There is a theorem: a matching m is maximum iff it has no augmenting path. This is a sufficient condition.

Note: matching, maximal matching, maxsimun matching, and augmenting paths are not only concepts of binary graphs. They are all concepts of graphs.

(4) Maximum match: match with the largest number of contained edges. HungaryAlgorithm[3]

(5) Perfect Match: all vertices are matched on the edge. [3]

(6) complete match: In the bipartite graph, all vertices in X have matched or all vertices in the Y set have matched accordingly. [3]

(7) Optimal Match: If G is a weighted bipartite graph, the perfect match between the weight value and the maximum value is called the best match. KM algorithm [3]

Refer:

[1] http://en.wikipedia.org/wiki/Matching_%28graph_theory%29#cite_note-Wes01-1

[2] http://en.wikipedia.org/wiki/Maximum_weight_matching

[3] http://www.cnblogs.com/one--world--one--dream/archive/2011/08/15/2139454.html

[4] http://www.cs.dartmouth.edu /~ AC/teach/CS105-Winter05/Notes/kavathekar-scribe.pdf