Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once. The cardinality of a matching is the number of matched edges. The weight of a matching is the sum of the weights of its edges.
- G (NetworkX graph) – Undirected graph
- maxcardinality (bool, optional) – If maxcardinality is True, compute the maximum-cardinality matching with maximum weight among all maximum-cardinality matchings.
mate – The matching is returned as a dictionary, mate, such that mate[v] == w if node v is matched to node w. Unmatched nodes do not occur as a key in mate.
If G has edges with ‘weight’ attribute the edge data are used as weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer computations. If floating point weights are used, the algorithm could return a slightly suboptimal matching due to numeric precision errors.
This method is based on the “blossom” method for finding augmenting paths and the “primal-dual” method for finding a matching of maximum weight, both methods invented by Jack Edmonds .
Bipartite graphs can also be matched using the functions present in
 “Efficient Algorithms for Finding Maximum Matching in Graphs”, Zvi Galil, ACM Computing Surveys, 1986.