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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 ...
In graph theory, a maximally matchable edge in a graph is an edge that is included in at least one maximum-cardinality matching in the graph. [1] An alternative term is allowed edge. [2] [3] A fundamental problem in matching theory is: given a graph G, find the set of all maximally matchable edges in G.
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 ...
Maximum cardinality matching is a fundamental problem in graph theory. [1] We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this ...
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 ...
In this graph, removing one vertex in the center produces three odd components, the three five-vertex lobes of the graph. Therefore, by the Tutte–Berge formula, it has at most (1−3+16)/2 = 7 edges in any matching.
For example, this 2-uniform hypergraph represents a graph with 4 vertices {1,2,3,4} and 3 edges: { {1,3}, {1,4}, {2,4} } By the above definition, a matching in a graph is a set M of edges, such that each two edges in M have an empty intersection.
In graph theory, Berge's theorem states that a matching M in a graph G is maximum (contains the largest possible number of edges) if and only if there is no augmenting path (a path that starts and ends on free (unmatched) vertices, and alternates between edges in and not in the matching) with M.