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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 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.
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 ...
A matching is a special case of a fractional matching in which all fractions are either 0 or 1. The size of a fractional matching is the sum of fractions of all hyperedges. The fractional matching number of a hypergraph H is the largest size of a fractional matching in H. It is often denoted by ν*(H). [3]
The matching is constructed by iteratively improving an initial empty matching along augmenting paths in the graph. 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.
Equivalently, it concerns the density of a different class of graphs, the locally linear graphs in which the neighborhood of every vertex is an induced matching. [4] Neither of these types of graph can have a quadratic number of edges, but constructions are known for graphs of this type with nearly-quadratic numbers of edges. [5]