# Matching (graph theory)

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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. Finding a matching in a bipartite graph can be treated as a network flow problem.

## Definitions

Given a graph G = (V,E), a matchingM in G is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share a common vertex.

A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched.

A maximal matching is a matching M of a graph G that is not a subset of any other matching. A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The following figure shows examples of maximal matchings (red) in three graphs.

A maximum matching (also known as maximum-cardinality matching [1] ) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number${\displaystyle \nu (G)}$ of a graph ${\displaystyle G}$ is the size of a maximum matching. Every maximum matching is maximal, but not every maximal matching is a maximum matching. The following figure shows examples of maximum matchings in the same three graphs.

A perfect matching is a matching that matches all vertices of the graph. That is, a matching is perfect if every vertex of the graph is incident to an edge of the matching. Every perfect matching is maximum and hence maximal. In some literature, the term complete matching is used. 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: ν(G) ρ(G) . 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) shows a near-perfect matching. If every vertex is unmatched by some near-perfect matching, then the graph is called factor-critical.

Given a matching M, an alternating path is a path that begins with an unmatched vertex [2] and whose edges belong alternately to the matching and not to the matching. An augmenting path is an alternating path that starts from and ends on free (unmatched) vertices. Berge's lemma states that a matching M is maximum if and only if there is no augmenting path with respect to M.

An induced matching is a matching that is the edge set of an induced subgraph. [3]

## Properties

In any graph without isolated vertices, the sum of the matching number and the edge covering number equals the number of vertices. [4] If there is a perfect matching, then both the matching number and the edge cover number are |V | / 2.

If A and B are two maximal matchings, then |A|  2|B| and |B|  2|A|. To see this, observe that each edge in B \ A can be adjacent to at most two edges in A \ B because A is a matching; moreover each edge in A \ B is adjacent to an edge in B \ A by maximality of B, hence

${\displaystyle |A\setminus B|\leq 2|B\setminus A|.}$

Further we deduce that

${\displaystyle |A|=|A\cap B|+|A\setminus B|\leq 2|B\cap A|+2|B\setminus A|=2|B|.}$

In particular, this shows that any maximal matching is a 2-approximation of a maximum matching and also a 2-approximation of a minimum maximal matching. This inequality is tight: for example, if G is a path with 3 edges and 4 vertices, the size of a minimum maximal matching is 1 and the size of a maximum matching is 2.

## Matching polynomials

A generating function of the number of k-edge matchings in a graph is called a matching polynomial. Let G be a graph and mk be the number of k-edge matchings. One matching polynomial of G is

${\displaystyle \sum _{k\geq 0}m_{k}x^{k}.}$

Another definition gives the matching polynomial as

${\displaystyle \sum _{k\geq 0}(-1)^{k}m_{k}x^{n-2k},}$

where n is the number of vertices in the graph. Each type has its uses; for more information see the article on matching polynomials.

## Algorithms and computational complexity

### Maximum-cardinality matching

A fundamental problem in combinatorial optimization is finding a maximum matching. This problem has various algorithms for different classes of graphs.

In an unweighted bipartite graph, the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time O(VE) time, and there are more efficient randomized algorithms, approximation algorithms, and algorithms for special classes of graphs such as bipartite planar graphs, as described in the main article.

### Maximum-weight matching

In a weighted bipartite graph, the optimization problem is to find a maximum-weight matching; a dual problem is to find a minimum-weight matching. This problem is often called maximum weighted bipartite matching, or the assignment problem . The Hungarian algorithm solves the assignment problem and it was one of the beginnings of combinatorial optimization algorithms. It uses a modified shortest path search in the augmenting path algorithm. If the Bellman–Ford algorithm is used for this step, the running time of the Hungarian algorithm becomes ${\displaystyle O(V^{2}E)}$, or the edge cost can be shifted with a potential to achieve ${\displaystyle O(V^{2}\log {V}+VE)}$ running time with the Dijkstra algorithm and Fibonacci heap. [5]

In a non-bipartite weighted graph, the problem of maximum weight matching can be solved in time ${\displaystyle O(V^{2}E)}$ using Edmonds' blossom algorithm.

### Maximal matchings

A maximal matching can be found with a simple greedy algorithm. A maximum matching is also a maximal matching, and hence it is possible to find a largest maximal matching in polynomial time. However, no polynomial-time algorithm is known for finding a minimum maximal matching, that is, a maximal matching that contains the smallest possible number of edges.

A maximal matching with k edges is an edge dominating set with k edges. Conversely, if we are given a minimum edge dominating set with k edges, we can construct a maximal matching with k edges in polynomial time. Therefore, the problem of finding a minimum maximal matching is essentially equal to the problem of finding a minimum edge dominating set. [6] Both of these two optimization problems are known to be NP-hard; the decision versions of these problems are classical examples of NP-complete problems. [7] Both problems can be approximated within factor 2 in polynomial time: simply find an arbitrary maximal matching M. [8]

### Counting problems

The number of matchings in a graph is known as the Hosoya index of the graph. It is #P-complete to compute this quantity, even for bipartite graphs. [9] It is also #P-complete to count perfect matchings, even in bipartite graphs, because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix. However, there exists a fully polynomial time randomized approximation scheme for counting the number of bipartite matchings. [10] A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm.

The number of perfect matchings in a complete graph Kn (with n even) is given by the double factorial (n  1)!!. [11] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers. [12]

### Finding all maximally-matchable edges

One of the basic problems in matching theory is to find in a given graph all edges that may be extended to a maximum matching in the graph (such edges are called maximally-matchable edges, or allowed edges). Algorithms for this problem include:

• For general graphs, a deterministic algorithm in time ${\displaystyle O(VE)}$ and a randomized algorithm in time ${\displaystyle {\tilde {O}}(V^{2.376})}$. [13] [14]
• For bipartite graphs, if a single maximum matching is found, a deterministic algorithm runs in time ${\displaystyle O(V+E)}$. [15]

### Online bipartite matching

The problem of developing an online algorithm for matching was first considered by Richard M. Karp, Umesh Vazirani, and Vijay Vazirani in 1990. [16]

In the online setting, nodes on one side of the bipartite graph arrive one at a time and must either be immediately matched to the other side of the graph or discarded. This is a natural generalization of the secretary problem and has applications to online ad auctions. The best online algorithm, for the unweighted maximization case with a random arrival model, attains a competitive ratio of 0.696. [17]

## Characterizations

Kőnig's theorem states that, in bipartite graphs, the maximum matching is equal in size to the minimum vertex cover. Via this result, the minimum vertex cover, maximum independent set, and maximum vertex biclique problems may be solved in polynomial time for bipartite graphs.

Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.

## Related Research Articles

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 =, a perfect matching in G is a subset M of E, such that every vertex in V is adjacent to exactly one edge in M.

The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows:

In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . The size of an independent set is the number of vertices it contains. Independent sets have also been called internally stable sets.

In the mathematical discipline of graph theory, a vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. The problem of finding a minimum vertex cover is a classical optimization problem in computer science and is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory.

In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph. Equivalently stated in symbolic terms an arbitrary graph is perfect if and only if for all we have .

In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

In graph theory, complete coloring is the opposite of harmonious coloring in the sense that it is a vertex coloring in which every pair of colors appears on at least one pair of adjacent vertices. Equivalently, a complete coloring is minimal in the sense that it cannot be transformed into a proper coloring with fewer colors by merging pairs of color classes. The achromatic number ψ(G) of a graph G is the maximum number of colors possible in any complete coloring of G.

In graph theory, a dominating set for a graph G = (VE) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G.

In graph theory, a domatic partition of a graph is a partition of into disjoint sets , ,..., such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph; here the dominating set consists of the yellow vertices, consists of the green vertices, and consists of the blue vertices.

In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.

In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In computer science, the minimum edge cover problem is the problem of finding an edge cover of minimum size. It is an optimization problem that belongs to the class of covering problems and can be solved in polynomial time.

In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (1931), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs.

In computer science, the Hopcroft–Karp algorithm is an algorithm that takes as input a bipartite graph and produces as output a maximum cardinality matching – a set of as many edges as possible with the property that no two edges share an endpoint. It runs in time in the worst case, where is set of edges in the graph, is set of vertices of the graph, and it is assumed that . In the case of dense graphs the time bound becomes , and for sparse random graphs it runs in near-linear time.

Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph , and the goal is to find a matching containing as many edges as possible, that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible.

The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network. A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated. The minimum cost flow problem is one of the most fundamental among all flow and circulation problems because most other such problems can be cast as a minimum cost flow problem and also that it can be solved efficiently using the network simplex algorithm.

In graph theory, a mathematical discipline, a factor-critical graph is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching.

In graph theory, boxicity is a graph invariant, introduced by Fred S. Roberts in 1969.

The blossom algorithm is an algorithm in graph theory for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, and published in 1965. Given a general graph G =, the algorithm finds a matching M such that each vertex in V is incident with at most one edge in M and |M| is maximized. The matching is constructed by iteratively improving an initial empty matching along augmenting paths in the graph. Unlike bipartite matching, the key new idea is that an odd-length cycle in the graph (blossom) is contracted to a single vertex, with the search continuing iteratively in the contracted graph.

In graph theory, a well-covered graph is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Equivalently, these are the graphs in which every maximal independent set has the same size. Well-covered graphs were defined and first studied by Plummer (1970).

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