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A maximal matching is a matching M of a graph G that is not a subset of any other matching. A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The following figure shows examples of maximal matchings (red) in three graphs. A maximum matching (also known as maximum-cardinality ...
The Hosoya index of a graph G, its number of matchings, is used in chemoinformatics as a structural descriptor of a molecular graph. It may be evaluated as m G (1) (Gutman 1991). The third type of matching polynomial was introduced by Farrell (1980) as a version of the "acyclic polynomial" used in chemistry.
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.
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.
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.
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. The naive reduction is to add n − r {\displaystyle n-r} new vertices to the smaller part and connect them to the larger part using edges of cost 0.
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 ]
Unlike bipartite matching, the key new idea is that an odd-length cycle in the graph (blossom) is contracted to a single vertex, with the search continuing iteratively in the contracted graph. The algorithm runs in time O (| E || V | 2 ) , where | E | is the number of edges of the graph and | V | is its number of vertices .