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The sum of weighted perfect matchings can also be computed by using the Tutte matrix for the adjacency matrix in the last step. Kuratowski's theorem states that 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).
The following are equivalent for a bipartite graph G = (X+Y, E): [13] G admits an X-perfect matching. G admits an X-perfect fractional matching. The implication follows directly from the fact that X-perfect matching is a special case of an X-perfect fractional matching, in which each weight is either 1 (if the edge is in the matching) or 0 (if ...
A graph can only contain a perfect matching when the graph has an even number of vertices. A near-perfect matching is one in which exactly one vertex is unmatched. Clearly, a graph can only contain a near-perfect matching when the graph has an odd number of vertices, and near-perfect matchings are maximum matchings. In the above figure, part (c ...
There is also a constant s which is at most the cardinality of a maximum matching in the graph. The goal is to find a minimum-cost matching of size exactly s. The most common case is the case in which the graph admits a one-sided-perfect matching (i.e., a matching of size r), and s=r. Unbalanced assignment can be reduced to a balanced assignment.
A perfect matching can only occur when the graph has an even number of vertices. 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.
In graph theory, the Dulmage–Mendelsohn decomposition is a partition of the vertices of a bipartite graph into subsets, with the property that two adjacent vertices belong to the same subset if and only if they are paired with each other in a perfect matching of the graph.
Kőnig had announced in 1914 and published in 1916 the results that every regular bipartite graph has a perfect matching, [11] and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it can be partitioned) equals its maximum degree [12] – the latter statement is known as Kőnig's ...
Any regular bipartite graph. [1] Hall's marriage theorem can be used to show that a k-regular bipartite graph contains a perfect matching. One can then remove the perfect matching to obtain a (k − 1)-regular bipartite graph, and apply the same reasoning repeatedly. Any complete graph with an even number of nodes (see below). [2]