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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 ...
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 ...
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 ...
The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts and . For example, the complete bipartite graph K 3,5 has degree sequence (,,), (,,,,). Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in ...
The Birkhoff polytope B n (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph , [1]) is the convex polytope in R N (where N = n 2) whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1.
A complete bipartite graph K m,n has a maximum matching of size min{m,n}. A complete bipartite graph K n,n has a proper n-edge-coloring corresponding to a Latin square. [14] Every complete bipartite graph is a modular graph: every triple of vertices has a median that belongs to shortest paths between each pair of vertices. [15]
The Dulmage-Mendelshon decomposition can be constructed as follows. [2] (it is attributed to [3] who in turn attribute it to [4]).Let G be a bipartite graph, M a maximum-cardinality matching in G, and V 0 the set of vertices of G unmatched by M (the "free vertices").
The sum of weighted perfect matchings can also be computed by using the Tutte matrix for the adjacency matrix in the last step. Kuratowski's theorem states that a finite graph is planar if and only if it contains no subgraph homeomorphic to K 5 (complete graph on five vertices) or K 3,3 (complete bipartite graph on two partitions of size three).