What is perfect bipartite matching?

The matching M is called perfect if for every v ∈ V , there is some e ∈ M which is incident on v. If a graph has a perfect matching, then clearly it must have an even number of vertices. Further- more, if a bipartite graph G = (L, R, E) has a perfect matching, then it must have |L| = |R|.

What is a perfect matching algorithm?

A perfect matching is a matching that matches all vertices of the graph. That is, a matching is perfect if every vertex of the graph is incident to an edge of the matching. Every perfect matching is maximum and hence maximal. A graph can only contain a perfect matching when the graph has an even number of vertices.

What is the maximum matching algorithm?

A common bipartite graph matching algorithm is the Hungarian maximum matching algorithm, which finds a maximum matching by finding augmenting paths. More formally, the algorithm works by attempting to build off of the current matching, M M M, aiming to find a larger matching via augmenting paths.

How do you find a matching bipartite graph?

Consider a bipartite graph G = (V,E) with bipartition (A, B) (V = A∪B). Let I = {X ⊆ A : there exists a matching M of G such that all vertices of X are matched}.

How to know if a bipartite graph has a perfect matching?

One possible way of \\fnding out if a given bipartite graph has a perfect matching is to use the above algorithm to \\fnd the maximum matching and checking if the size of the matching equals the number of nodes in each partition. There is another way of determining this, using Hall’s Theorem. 2

What is the maximum size of a bipartite matching?

Maximum Bipartite Matching. A matching in a Bipartite Graph is a set of the edges chosen in such a way that no two edges share an endpoint. A maximum matching is a matching of maximum size (maximum number of edges).

Which is an example of a bipartite matching problem?

There are many real world problems that can be formed as Bipartite Matching. For example, consider the following problem: There are M job applicants and N jobs. Each applicant has a subset of jobs that he/she is interested in.

Which is the correct theorem for a bipartite graph?

To argue that this approach is correct, we must prove the following theorem: Theorem 4.1 For a given bipartite graph G, a matching M is maximum if and only if G has no augmenting paths with respect to M. Proof: ()) We prove this by contrapositive, i.e., by showing that if G has an augmenting path, then M is not a maximum matching.