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Maximum cardinality matching is a fundamental problem in graph theory. [1] We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this ...
A maximum matching (also known as maximum-cardinality matching [2]) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number of a graph G is the size of a maximum matching. Every maximum matching is maximal, but not every maximal matching is a maximum matching.
In computer science, the Hopcroft–Karp algorithm (sometimes more accurately called the Hopcroft–Karp–Karzanov algorithm) [1] is an algorithm that takes a bipartite graph as input and produces a maximum-cardinality matching as output — a set of as many edges as possible with the property that no two edges share an endpoint.
The problem of finding a maximum-cardinality matching in a hypergraph, thus calculating (), is NP-hard even for 3-uniform hypergraphs (see 3-dimensional matching). This is in contrast to the case of simple (2-uniform) graphs in which computing a maximum-cardinality matching can be done in polynomial time.
Let G = (V,E) be a graph, where V are the vertices and E are the edges. A matching in G is a subset M of E, such that each vertex in V is adjacent to at most a single edge in M. A maximum matching is a matching of maximum cardinality. An edge e in E is called maximally matchable (or allowed) if there exists a maximum matching M that contains e.
The fifth corner (1/2,1/2,1/2) does not represent a matching - it represents a fractional matching in which each edge is "half in, half out". Note that this is the largest fractional matching in this graph - its weight is 3/2, in contrast to the three integral matchings whose size is only 1. As another example, in the 4-cycle there are 4 edges.
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
The Dulmage-Mendelshon decomposition can be constructed as follows. [2] (it is attributed to [3] who in turn attribute it to [4]).Let G be a bipartite graph, M a maximum-cardinality matching in G, and V 0 the set of vertices of G unmatched by M (the "free vertices").