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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 ...
Graph coloring [2] [3]: GT4 Graph homomorphism problem [3]: GT52 Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph.
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = { 1 , 2 , … , n } {\displaystyle V=\{1,2,\dots ,n\}} then the Tutte matrix is an n -by- n matrix A with entries
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 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.
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 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 ...
This alternating cycle may be used to modify the initial perfect matching to produce a new matching containing edge x,y. An edge x,y of the graph G belongs to all perfect matchings of G, if and only if x and y are the only members of their set in the decomposition. Such an edge exists if and only if the matching preclusion number of the graph ...