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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, the size of a maximum matching is no larger than the size of a minimum edge cover: () . A graph can only contain a perfect matching when the graph has an even ...
The Fisher–Kasteleyn–Temperley (FKT) algorithm, named after Michael Fisher, Pieter Kasteleyn, and Neville Temperley, counts the number of perfect matchings in a planar graph in polynomial time. This same task is #P-complete for general graphs. For matchings that are not required to be perfect, counting them remains #P-complete even for ...
The case of exact graph matching is known as the graph isomorphism problem. [1] The problem of exact matching of a graph to a part of another graph is called subgraph isomorphism problem. Inexact graph matching refers to matching problems when exact matching is impossible, e.g., when the number of vertices in the two graphs are different. In ...
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 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.
For instance, if G = K m,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial L n α (x) by the identity: , =! (). If G is the complete graph K n, then M G (x) is an Hermite polynomial:
In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is the assignment problem , in which the input is restricted to be a bipartite graph , and the matching constrained to be have cardinality that of the ...
Matching (graph theory) – matching between different vertices of the graph; usually unrelated to preference-ordering. Envy-free matching – a relaxation of stable matching for many-to-one matching problems; Rainbow matching for edge colored graphs; Stable matching polytope; Lattice of stable matchings