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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 as a network flow problem.
A complete bipartite graph with m = 5 and n = 3 The Heawood graph is bipartite.. In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in .
A special case of the maximum weight matching problem is the assignment problem, in which the graph is a bipartite graph and the matching must have cardinality equal to that of the smaller of the two partitions. Another special case is the problem of finding a maximum cardinality matching on an unweighted graph: this corresponds to the case ...
More efficient algorithms exist for special kinds of bipartite graphs: For sparse bipartite graphs, the maximum matching problem can be solved in ~ (/) with Madry's algorithm based on electric flows. [3] For planar bipartite graphs, the problem can be solved in time O(n log 3 n) where n is the number of vertices, by reducing the problem to ...
The perfect matching polytope of a graph G, denoted PMP(G), is a polytope whose corners are the incidence vectors of the integral perfect matchings in G. [1]: 274 Obviously, PMP(G) is contained in MP(G); In fact, PMP(G) is the face of MP(G) determined by the equality:
In computer science, the Hopcroft–Karp algorithm (sometimes more accurately called the Hopcroft–Karp–Karzanov algorithm) [1] is an algorithm that takes a bipartite graph as input and produces a maximum-cardinality matching as output — a set of as many edges as possible with the property that no two edges share an endpoint.
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
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]