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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 ...
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 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.
A matching M is called perfect if every vertex v in V is contained in exactly one hyperedge of M. This is the natural extension of the notion of perfect matching in a graph. A fractional matching M is called perfect if for every vertex v in V, the sum of fractions of hyperedges in M containing v is exactly 1.
In a cubic graph with a perfect matching, the edges that are not in the perfect matching form a 2-factor. By orienting the 2-factor, the edges of the perfect matching can be extended to paths of length three, say by taking the outward-oriented edges. This shows that every cubic, bridgeless graph decomposes into edge-disjoint paths of length ...
The minimum degree of a graph, often denoted by deg(G) or δ(v), is the minimum of deg(v) over all vertices v in V. A matching in a graph is a set of edges such that each vertex is adjacent to at most one edge; a perfect matching is a matching in which each vertex is adjacent to exactly one edge. A perfect matching does not always exist, and ...
In graph theory, the matching polytope of a given graph is a geometric object representing the possible matchings in the graph. It is a convex polytope each of whose corners corresponds to a matching. It has great theoretical importance in the theory of matching. [1]: 273–285
The matching complex of a hypergraph is exactly the independence complex of its line graph, denoted L(H). This is a graph in which the vertices are the edges of H, and two such vertices are connected iff their corresponding edges intersect in H. Therefore, the above theorem implies: