In graph theory, a **matching** is a set of edges in a graph such that no two edges share a common vertex. In other words, it is a way of pairing the vertices of a graph such that each vertex is included in at most one edge of the matching. Here's a more detailed explanation: 1. **Vertices and Edges**: A graph consists of a set of vertices (nodes) and a set of edges (connections between pairs of vertices).
Stable matching is a concept primarily found in the field of game theory and economics, particularly in the context of matching markets. It refers to a situation where members of two different sets (commonly referred to as "agents") are paired in a way that no two individuals would prefer to be matched with each other over their current partners. This concept was popularized by the Gale-Shapley algorithm, which was introduced in their seminal paper "College Admissions and the Stability of Marriage" in 1962.
The Assignment Problem is a fundamental problem in combinatorial optimization that involves assigning a set of resources to a set of tasks in such a way that the total cost is minimized (or, in some cases, maximized). It can be represented mathematically and is commonly solved using various optimization techniques.
Berge's theorem is a foundational result in combinatorial optimization and graph theory, specifically relating to bipartite graphs. The theorem provides a characterization of maximum matchings in bipartite graphs and links it to the concept of "augmenting paths.
A chord diagram is a graphical representation used to visualize relationships or connections between different entities within a circular layout. In mathematics and data visualization, it typically consists of a circle where each entity (or node) is represented by a point on the circumference of the circle. Chords are drawn between these points to represent the relationships or interactions between the entities. **Key Features of a Chord Diagram:** 1.
A **claw-free graph** is a specific type of graph in graph theory that does not contain a particular induced subgraph known as a "claw." A claw is defined as a complete bipartite graph \( K_{1,3} \), which can be visualized as a star with one central vertex connected to three other vertices. In other words, a claw consists of one vertex (the center) connected to three other vertices (the leaves), with no other connections among them.
A *factor-critical graph* is a type of graph in which the removal of any single vertex results in a graph that has a perfect matching. In other words, a graph \( G \) is called factor-critical if for every vertex \( v \) in \( G \), the graph \( G - v \) (the graph obtained by removing vertex \( v \) and its incident edges) has a perfect matching.
Fractional matching is a concept primarily used in graph theory and combinatorial optimization. It generalizes the idea of matching in bipartite graphs, allowing for a scenario where matches can be made in fractions rather than whole numbers, which means that connections between pairs of vertices can be partially utilized. ### Key Concepts: 1. **Matching**: In a traditional matching context, a matching in a graph is a set of edges such that no two edges share a vertex.
The House Allocation Problem is a classic problem in economics and game theory that deals with the distribution of a fixed number of houses (or resources) among a group of individuals (or agents) based on their preferences. Each individual typically has their own rankings of the available houses, and the objective is to allocate the houses in a way that is fair and efficient.
The Hungarian algorithm is a combinatorial optimization algorithm used to solve the assignment problem, which involves finding the optimal way to assign a set of tasks to a set of agents such that the total cost is minimized. The assignment problem can be represented mathematically using a cost matrix, where each element represents the cost of assigning a particular task to a particular agent.
Induced matching is a concept used primarily in the fields of psychiatry and social sciences, particularly in the context of observational studies and nonrandomized trials. The idea behind induced matching is to reduce bias in estimates of treatment effects by matching subjects in a way that accounts for certain covariates that could influence both treatment assignment and outcomes. In induced matching, subjects who receive a particular treatment are matched with subjects who do not, on the basis of observable characteristics.
Kőnig's theorem is a fundamental result in graph theory, specifically in the area of bipartite graphs and matching. It states that in any bipartite graph, the size of the maximum matching (the largest set of edges that do not share a vertex) is equal to the size of the minimum vertex cover (the smallest set of vertices such that every edge in the graph is incident to at least one vertex from this set).
In the context of hypergraphs, a **matching** refers to a set of edges such that no two edges share a common vertex. A hypergraph is a generalization of a graph where an edge can connect any number of vertices, not just two.
The matching polytope is a mathematical structure associated with the graph theory concept of matchings in bipartite graphs. In general, a matching in a graph is a set of edges such that no two edges share a common vertex. The matching polytope is particularly defined for bipartite graphs, although it can extend to general graphs.
Matching preclusion is a concept primarily used in the field of graph theory and combinatorial optimization, particularly in the study of matchings in bipartite graphs. The central idea of matching preclusion is to identify and analyze conditions under which certain edges or vertices in a graph can be "excluded" from being part of any maximum matching.
In the context of graph theory, particularly when discussing matchings in bipartite graphs, a **maximally matchable edge** refers to an edge in a matching that cannot be included in a larger matching without violating the properties of disjointness. ### Key Concepts: 1. **Matching**: A matching in a graph is a set of edges without common vertices. A perfect matching is a matching that covers every vertex of the graph. 2. **Maximal Matching vs.
Maximum cardinality matching is a concept in graph theory referring to a matching (a set of edges without common vertices) that includes the maximum number of edges possible. In a simple undirected graph, a matching pairs up vertices such that no two edges share a vertex. ### Key Points: 1. **Matching**: A matching in a graph is a set of edges where no two edges share a vertex.
In graph theory, a **perfect matching** (or complete matching) is a specific type of matching in a graph. A matching is defined as a set of edges without common vertices. In the case of a perfect matching, each vertex of the graph is included in exactly one edge of the matching.
Perfect matching in high-degree hypergraphs is an extension of the concept of matching from standard graphs to hypergraphs, which are generalizations where edges can connect more than two vertices. Specifically, a hypergraph \( H \) consists of a set of vertices \( V \) and a set of edges \( E \), where each edge \( e \in E \) is a subset of \( V \) with more than two vertices.
Petersen's theorem refers to a specific result in graph theory related to the structure of graphs. It states that every cubic vertex-transitive graph that is not bipartite contains a Hamiltonian cycle. A graph is cubic if every vertex has degree 3 (i.e., each vertex is connected to exactly three other vertices).
Pfaffian orientation is a concept in graph theory, particularly related to the study of oriented graphs and the enumeration of perfect matchings. It's most commonly associated with bipartite graphs and has a connection to the determinant of certain matrices. ### Key Concepts: 1. **Directed Graphs**: In graph theory, a directed graph (or digraph) consists of vertices connected by edges, where each edge has a direction.
Priority matching is a concept used in various fields, including finance, human resources, and computer science, among others. The specific meaning can vary depending on the context, but generally, priority matching refers to pairing or aligning two or more parties or items based on a set of prioritized criteria. Here are a couple of contexts in which the term can be applied: 1. **Finance and Trading**: In financial markets, priority matching refers to the process of executing trades based on the prioritization of orders.
Rank-maximal allocation is a concept that arises in the context of resource allocation problems, particularly in matching markets and auctions. The idea is to allocate resources (such as goods or services) to agents (such as individuals or organizations) in a way that maximizes the rank of the allocated outcomes according to each agent's preferences. In simpler terms, rank-maximal allocation attempts to ensure that each agent receives an allocation that is as high as possible on their personal preference list.
The Ruzsa–Szemerédi problem is a question in the field of combinatorial number theory, particularly concerning sets of integers and their structure. It was posed by Hungarian mathematicians Imre Ruzsa and Endre Szemerédi. The problem revolves around the concept of progressions in subsets of integers. Specifically, it asks how large a subset of integers can be if it avoids certain arithmetic progressions of a given length.
In graph theory, "saturation" refers to the concept of a saturated graph or a saturated set of edges relative to a given property. The term can have specific meanings depending on the context in which it is used, but generally, it involves the idea of maximizing certain characteristics or properties of a graph while avoiding others.
A skew-symmetric graph, in the context of graph theory, refers to a special type of directed graph (digraph) that exhibits certain symmetrical properties in its edges.
The Top Trading Cycle (TTC) is a notable algorithm used in the field of resource allocation and matching theory. It was primarily developed by economists to allocate resources or items efficiently among a group of agents based on their preferences. The basic idea of the Top Trading Cycle algorithm is as follows: 1. **Initial Setup**: Each participant (agent) has a list of preferences, indicating which items they would like to receive.
The Tutte theorem, also known as the Tutte-Berge formula, is a fundamental result in graph theory concerning perfect matchings in bipartite and general graphs. The theorem provides necessary and sufficient conditions for the existence of a perfect matching in a graph.
The Tutte–Berge formula is a fundamental result in graph theory that relates to the maximum size of a matching in a bipartite graph. It provides a way to determine whether a particular matching covers all vertices of the graph.
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