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The number of perfect matchings in a complete graph K n (with n even) is given by the double factorial (n − 1)!!. [13] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers. [14] The number of perfect matchings in a graph is also known as the hafnian of its adjacency ...
However, counting the number of perfect matchings, even in bipartite graphs, is #P-complete. This is because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix.
a finite graph is planar if and only if it contains no subgraph homeomorphic to K 5 (complete graph on five vertices) or K 3,3 (complete bipartite graph on two partitions of size three). Vijay Vazirani generalized the FKT algorithm to graphs that do not contain a subgraph homeomorphic to K 3,3 . [ 11 ]
The assignment problem consists of finding, in a weighted bipartite graph, a matching of maximum size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment, and the graph-theoretic version is called minimum-cost perfect matching.
In a uniformly-random instance of the stable marriage problem with n men and n women, the average number of stable matchings is asymptotically . [6] In a stable marriage instance chosen to maximize the number of different stable matchings, this number is an exponential function of n . [ 7 ]
The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts and . For example, the complete bipartite graph K 3,5 has degree sequence (,,), (,,,,). Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in ...
In graph theory, a Pfaffian orientation of an undirected graph assigns a direction to each edge, so that certain cycles (the "even central cycles") have an odd number of edges in each direction. When a graph has a Pfaffian orientation, the orientation can be used to count the perfect matchings of the graph.
One application of the Edmonds matrix of a bipartite graph is that the graph admits a perfect matching if and only if the polynomial det(A ij) in the x ij is not identically zero. Furthermore, the number of perfect matchings is equal to the number of monomials in the polynomial det(A), and is also equal to the permanent of .