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In the mathematical discipline of graph theory, a matching or independent edge 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. Bipartite matching is a special case of a network flow problem. == 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. A vertex is matched (or saturated) if it is an endpoint of one of the edges 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''. The following figure shows examples of maximal matchings (red) in three graphs. :File:Maximal-matching.svg A maximum matching (also known as maximum-cardinality matching〔Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5.〕) 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 is maximal, but not every maximal matching is a maximum matching. The following figure shows examples of maximum matchings in the same three graphs. :File:Maximum-matching-labels.svg A perfect matching (a.k.a. 1-factor) is a matching which matches all vertices of the graph. That is, every vertex of the graph is incident to exactly one edge of the matching. Figure (b) above is an example of a perfect matching. Every perfect matching is maximum and hence maximal. In some literature, the term complete matching is used. In the above figure, only part (b) shows a perfect matching. A perfect matching is also a minimum-size edge cover. Thus, , that is, the size of a maximum matching is no larger than the size of a minimum edge cover. A near-perfect matching is one in which exactly one vertex is unmatched. This can only occur when the graph has an odd number of vertices, and such a matching must be maximum. In the above figure, part (c) shows a near-perfect matching. If, for every vertex in a graph, there is a near-perfect matching that omits only that vertex, the graph is also called factor-critical. Given a matching ''M'', * an alternating path is a that begins with an unmatched vertex and is a 〔http://diestel-graph-theory.com/basic.html〕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.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Matching (graph theory)」の詳細全文を読む スポンサード リンク
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