|
Dheeraj Gedam This article is about mathematical matchings. For matching in the field of electronics, see: Impedance matching and Impedance bridging Impedance matching is the practice of attempting to make the output impedance of a source equal to the input impedance of the load to which it is ultimately connected, usually in order to maximise the power transfer and minimise reflections from the load. ...
For the amplifier configuration, see bridged amplifier. ...
In the mathematical discipline of graph theory a matching or edge independent set in a graph is a set of edges without common vertices. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ...
A graph diagram of a graph with 6 vertices and 7 edges. ...
In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ...
Definition Given a graph G = (V,E) a matching M in G is a set of pairwise non-adjacent edges. A graph with 6 vertices (nodes) and 7 edges. ...
We say that a vertex is matched if it is incident to an edge in the matching. Otherwise the vertex is unmatched.
A maximum matching is a matching that is as big (has as many edges) as possible. There may be many maximum matchings.
An alternating path is a path in which the edges belong alternatively to the matching and not to the matching.
An augmenting path is an alternating path that starts from and ends to free (unmatched) vertices.
A matching is maximum if and only if it does not contain any augmenting path.
The matching number of a graph is the size of a maximum matching.
A perfect matching is a matching which covers all vertices of the graph. That is, every vertex of the graph is incident to exactly one edge of the matching. A graph with 6 vertices (nodes) and 7 edges. ...
Examples In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. ...
Notes Some people also allow graphs with an odd number of vertices to have a "perfect" matching. These matchings have exactly one unmatched vertex. In mathematics, any integer (whole number) is either even or odd. ...
The marriage theorem provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs. In mathematics, the marriage theorem (1935), usually credited to mathematician Philip Hall, is a combinatorial result that gives the condition allowing the selection of a distinct element from each of a collection of subsets. ...
In the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of graphs with perfect matchings. ...
Hopcroft and Karp proposed an algorithm for computing maximum matchings on bipartite graphs that runs in  where n is the number of vertices and m is the number of edges.
There is a polynomial time algorithm (sometimes called the Hungarian algorithm) which, given a bipartite graph G, determines if G has a perfect matching and, if it has, finds a perfect matching. There is also a polynomial time algorithm to find a maximum matching in a graph that is not bipartite; it is due to Edmonds, is called the paths, trees, and flowers method, and uses bidirected edges. In computational complexity theory, Polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ...
In graph theory, Munkres assignment algorithm or the Hungarian algorithm is an algorithm which solves the assignment problem in polynomial time. ...
In the mathematical domain of graph theory, a bidirected graph is a graph in which each edge is given an independent orientation (or direction, or arrow) at each end. ...
A related problem is, given a graph G, to determine the number of perfect matchings in G. This problem is #P Complete. For bipartite graphs, it can be approximately solved in polynomial time. That is, for any ε>0, there is a probabilistic polynomial time algorithm that determines, with high probability, the number of perfect matchings M within an error of at most εM. The title given to this article is incorrect due to technical limitations. ...
In the mathematical field of graph theory, a bipartite graph is a special graph where the set of vertices can be divided into two disjoint sets with two vertices of the same set never sharing an edge. ...
In computer science, approximation algorithms are an approach to attacking NP-hard optimization problems. ...
A randomized algorithm is an algorithm which is allowed to flip a truly random coin. ...
The word probability derives from the Latin probare (to prove, or to test). ...
Properties In mathematics and computer science, graph theory studies the properties of graphs. ...
In the mathematical discipline of graph theory a covering for a graph is a set of vertices (or edges) so that the elements of the set are close (adjacent) to all edges (or vertices) of the graph. ...
Tibor Gallai (1912–1992) was a Hungarian mathematician. ...
See also In graph theory, an independent, or stable, set in a graph G, which contains vertices V, is a set of vertices V (a subset of V) such that for every two vertices in V, there is no edge connecting the two. ...
In Graph theory, The Dulmage-Mendelson decomposition is a method used to create an maximal matching on a bipartite graph. ...
In graph theory, a network flow is an assignment of flow to the edges of a directed graph (called a flow network in this case) where each edge has a capacity, such that the amount of flow along an edge does not exceed its capacity. ...
References - Gallai, Tibor. "Über extreme Punkt- und Kantenmengen." Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 2, 133-138, 1959.
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 26.3: Maximum bipartite matching, pp.664–669.
|
Feeds |
Contact