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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 = (V, E), a perfect matching in G is a subset M of edge set E, such that every vertex in the vertex set V is adjacent to exactly one edge in M. A perfect matching is also called a 1-factor; see Graph factorization ...
A cubic (but not bridgeless) graph with no perfect matching, showing that the bridgeless condition in Petersen's theorem cannot be omitted. In the mathematical discipline of graph theory, Petersen's theorem, named after Julius Petersen, is one of the earliest results in graph theory and can be stated as follows: Petersen's Theorem.
Matching (graph theory) In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. [1] In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated ...
Stable marriage problem. In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the elements of ...
A fractional matching in a graph is an assignment of non-negative weights to each edge, such that the sum of weights adjacent to each vertex is at most 1. A fractional matching is X-perfect if the sum of weights adjacent to each vertex is exactly 1. The following are equivalent for a bipartite graph G = (X+Y, E): [13] G admits an X-perfect ...
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.
Tutte–Berge formula. 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. In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the ...
Creating a matching Using a shortcut heuristic on the graph created by the matching above. The algorithm of Christofides and Serdyukov follows a similar outline but combines the minimum spanning tree with a solution of another problem, minimum-weight perfect matching. This gives a TSP tour which is at most 1.5 times the optimal.