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In particular, computing the matching polynomial on n-vertex graphs of treewidth k is fixed-parameter tractable: there exists an algorithm whose running time, for any fixed constant k, is a polynomial in n with an exponent that does not depend on k (Courcelle, Makowsky & Rotics 2001).
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.
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, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G with edges E and vertices V, a perfect matching in G is a subset M of E, such that every vertex in V is adjacent to exactly one edge in M. The adjacency matrix of a perfect matching is a symmetric permutation matrix.
The hafnian of an adjacency matrix of a graph is the number of perfect matchings (also known as 1-factors) in the graph. This is because a partition of { 1 , 2 , … , 2 n } {\displaystyle \{1,2,\dots ,2n\}} into subsets of size 2 can also be thought of as a perfect matching in the complete graph K 2 n {\displaystyle K_{2n}} .
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.
Graphs are commonly used to encode structural information in many fields, including computer vision and pattern recognition, and graph matching, i.e., identification of similarities between graphs, is an important tools in these areas. In these areas graph isomorphism problem is known as the exact graph matching. [47]