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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 ...
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 ...
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.
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.
A balanced tripartite graph with the unique triangle property can be made into a partitioned bipartite graph by removing one of its three subsets of vertices, and making an induced matching on the neighbors of each removed vertex. To convert a graph with a unique triangle per edge into a triple system, let the triples be the triangles of the graph.
Given a graph , its Gallai–Edmonds decomposition consists of three disjoint sets of vertices, (), (), and (), whose union is (): the set of all vertices of .First, the vertices of are divided into essential vertices (vertices which are covered by every maximum matching in ) and inessential vertices (vertices which are left uncovered by at least one maximum matching in ).
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.
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