# Category: Matching (graph theory)

Priority matching
In graph theory, a priority matching (also called: maximum priority matching) is a matching that maximizes the number of high-priority vertices that participate in the matching. Formally, we are given
Saturation (graph theory)
Given a graph , another graph is -saturated if does not contain a (not necessarily induced) copy of , but adding any edge to it does. The function is the minimum number of edges an -saturated graph on
Tutte theorem
In the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of finite graphs with perfect matchings. It is a generalization of Hall's marr
Fractional matching
In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices.
Hopcroft–Karp algorithm
In computer science, the Hopcroft–Karp algorithm (sometimes more accurately called the Hopcroft–Karp–Karzanov algorithm) is an algorithm that takes a bipartite graph as input and produces a maximum ca
Hafnian
In mathematics, the hafnian of an adjacency matrix of a graph is the number of perfect matchings in the graph. It was so named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhag
House allocation problem
In economics and computer science, the house allocation problem is the problem of assigning objects to people with different preferences, such that each person receives exactly one object. The name "h
Chord diagram (mathematics)
In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by
Matching polytope
In graph theory, the matching polytope of a given graph is a geometric object representing the possible matchings in the graph. It is a convex polytope each of whose corners corresponds to a matching.
Perfect matching
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 t
Pfaffian orientation
In graph theory, a Pfaffian orientation of an undirected graph assigns a direction to each edge, so that certain cycles (the "even central cycles") have an odd number of edges in each direction. When
Hall-type theorems for hypergraphs
In the mathematical field of graph theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by Ofra Kessl
Petersen's theorem
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. Every
Matching in hypergraphs
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph.
Matching preclusion
In graph theory, a branch of mathematics, the matching preclusion number of a graph G (denoted mp(G)) is the minimum number of edges whose deletion results in the destruction of a perfect matching or
Ruzsa–Szemerédi problem
In combinatorial mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs to a unique triangle.Eq
Skew-symmetric graph
In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism
1-factor
No description available.
Hall's marriage theorem
In mathematics, Hall's marriage theorem, proved by Philip Hall, is a theorem with two equivalent formulations: * The combinatorial formulation deals with a collection of finite sets. It gives a neces
Maximum cardinality matching
Maximum cardinality matching is a fundamental problem in graph theory.We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality su
Top trading cycle (TTC) is an algorithm for trading indivisible items without using money. It was developed by David Gale and published by Herbert Scarf and Lloyd Shapley.
Quantum dimer models
Quantum dimer models were introduced to model the physics of resonating valence bond (RVB) states in . The only degrees of freedom retained from the motivating spin systems are the valence bonds, repr
Tutte matrix
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If
Claw-free graph
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K1,3 (that is, a star gr
Matching polynomial
In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a
Partition matroid
In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different cat
Dulmage–Mendelsohn decomposition
In graph theory, the Dulmage–Mendelsohn decomposition is a partition of the vertices of a bipartite graph into subsets, with the property that two adjacent vertices belong to the same subset if and on
Perfect matching in high-degree hypergraphs
In graph theory, perfect matching in high-degree hypergraphs is a research avenue trying to find sufficient conditions for existence of a perfect matching in a hypergraph, based only on the degree of
Hungarian algorithm
The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal–dual methods. It was developed and published in
Tutte–Berge formula
In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte theorem on perfect matchings,
Maximally-matchable edge
In graph theory, a maximally-matchable edge in a graph is an edge that is included in at least one maximum-cardinality matching in the graph. An alternative term is allowed edge. A fundamental problem
Assignment problem
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any
Hosoya index
The Hosoya index, also known as the Z index, of a graph is the total number of matchings in it. The Hosoya index is always at least one, because the empty set of edges is counted as a matching for thi
Rank-maximal allocation
Rank-maximal (RM) allocation is a rule for fair division of indivisible items. Suppose we have to allocate some items among people. Each person can rank the items from best to worst. The RM rule says
Transportation theory (mathematics)
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mat
Gallai–Edmonds decomposition
In graph theory, the Gallai–Edmonds decomposition is a partition of the vertices of a graph into three subsets which provides information on the structure of maximum matchings in the graph. Tibor Gall
Blossom algorithm
In graph theory, the blossom algorithm is an algorithm for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, and published in 1965. Given a general graph G
Induced matching
In graph theory, an induced matching or strong matching is a subset of the edges of an undirected graph that do not share any vertices (it is a matching) and includes every edge connecting any two ver
Weapon target assignment problem
The weapon target assignment problem (WTA) is a class of combinatorial optimization problems present in the fields of optimization and operations research. It consists of finding an optimal assignment
Factor-critical graph
In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph) is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching. (A perfect mat
3-dimensional matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyp
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. Finding a matching in a bipartite graph can be trea
Berge's theorem
In graph theory, Berge's theorem states that a matching M in a graph G is maximum (contains the largest possible number of edges) if and only if there is no augmenting path (a path that starts and end
Secretary problem
The secretary problem demonstrates a scenario involving optimal stopping theory that is studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as t
Kőnig's theorem (graph theory)
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
Domino tiling
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a
Telephone number (mathematics)
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person telephone calls. These numbers also descri