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.

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• 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 $x\; \backslash in\; X$ 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 $O(\backslash sqrt\{V\}\; E)$ time.