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Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: [1] In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are ...
In the above figure, only part (b) shows a perfect matching. A perfect matching is also a minimum-size edge cover. Thus, the size of a maximum matching is no larger than the size of a minimum edge cover: () . 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 ...
As shown by Mulmuley, Vazirani and Vazirani, [8] the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph. Using the isolation lemma, a minimum weight perfect matching in a graph can be found with probability at least 1 ⁄ 2.
The minimum-weight perfect matching can have no larger weight, so w(M) ≤ w(C)/2. Adding the weights of T and M gives the weight of the Euler tour, at most 3w(C)/2. Thanks to the triangle inequality, even though the Euler tour might revisit vertices, shortcutting does not increase the weight, so the weight of the output is also at most 3w(C)/2 ...
The following figure shows examples of minimum edge coverings (again, the set C is marked with red). Note that the figure on the right is not only an edge cover but also a matching. In particular, it is a perfect matching: a matching M in which every vertex is incident with exactly one edge in M. A perfect matching (if it exists) is always a ...
Given a bipartite graph G = (A ∪ B, E), the goal is to find the maximum cardinality matching in G that has minimum cost. Let w: E → R be a weight function on the edges of E. The minimum weight bipartite matching problem or assignment problem is to find a perfect matching M ⊆ E whose total weight is minimized. The idea is to reduce this ...
Minimum maximal independent set a.k.a. minimum independent dominating set [4] NP-complete special cases include the minimum maximal matching problem, [3]: GT10 which is essentially equal to the edge dominating set problem (see above). Metric dimension of a graph [3]: GT61 Metric k-center; Minimum degree spanning tree; Minimum k-cut
A minimum T-join can be obtained by constructing a complete graph on the vertices of T, with edges that represent shortest paths in the given input graph, and then finding a minimum weight perfect matching in this complete graph. The edges of this matching represent paths in the original graph, whose union forms the desired T-join.