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,
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).
In a weighted bipartite graph, each edge has an associated value. A maximum weighted bipartite matching is defined as a perfect matching where the sum of the values of the edges in the matching have a maximal value. If the graph is not complete bipartite, missing edges are inserted with value zero. Finding such a matching is known as the assignment problem. It can be solved by using a modified shortest path search in the augmenting path algorithm. If the Bellman-Ford algorithm is used, the running time becomes , or the edge cost can be shifted with a potential to achieve running time with the Dijkstra algorithm and Fibonacci heap. The remarkable Hungarian algorithm solves the assignment problem and it was one of the starting point of the combinatorial optimization. The original approach of this algorithm need running time, but it could be improved to time with extensive use of priority queues.
König's theorem states that, in bipartite graphs, the maximum matching is equal in size to the minimum vertex cover. Via this result, the minimum vertex cover problem and maximum independent set problem may be solved in polynomial time for bipartite graphs.
A related problem is, given a graph G, to determine the number of perfect matchings in G. This problem is #P Complete (see Permanent). However, a remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time. Also, for bipartite graphs, the problem can be approximately solved in polynomial time. That is, for any ε>0, there is a probabilistic polynomial time algorithm that determines, with high probability, the number of perfect matchings M within an error of at most εM.
Matching problem is a special case of a more general problem called f-Factor problem. A graph G=(V,E) has an f-Factor if and only if, G has a subgraph G', where the degree of each node v in G' is f(v). If f(v)=1 for all vertices, f-Factor problem reduces to matching problem. Hence matching problem is also referred to as 1-Factor problem.
In chemistry, a Kekulé structure of an aromatic compound consists of a perfect matching of its carbon skeleton, showing the locations of double bonds in the chemical structure. These structures are named after Friedrich August Kekulé von Stradonitz, who showed that benzene (in graph theoretical terms, a 6-vertex cycle) can be given such a structure.
In 1971, Haruo Hosoya defined topological index (a graph invariant) as the total number of matchings of a graph plus 1. The Hosoya index is often used in computer chemistry investigations for organic compounds.