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Deficiency (graph Theory)
Deficiency is a concept in graph theory that is used to refine various theorems related to perfect matching in graphs, such as Hall's marriage theorem. This was first studied by Øystein Ore. A related property is surplus. Definition of deficiency Let be a graph, and let ''U'' be an independent set of vertices, that is, ''U'' is a subset of ''V'' in which no two vertices are connected by an edge. Let denotes the set of neighbors of ''U'', which is formed by all vertices from 'V' that are connected by an edge to one or more vertices of ''U''. The deficiency of the set ''U'' is defined by: :\mathrm_G(U) := , U, - , N_G(U), Suppose ''G'' is a bipartite graph, with bipartition ''V'' = ''X'' ∪ ''Y''. The deficiency of ''G'' with respect to one of its parts (say ''X''), is the maximum deficiency of a subset of ''X'': :\mathrm(G;X) := \max_ \mathrm_G(U) Sometimes this quantity is called the critical difference of ''G''. Note that defG of the empty subset is 0, so def(''G;''X) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 , a perfect matching in is a subset of edge set , such that every vertex in the vertex set is adjacent to exactly one edge in . A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cover. If there is a perfect matching, then both the matching number and the edge cover number equal . A perfect matching can only occur when the graph has an even num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall's Marriage Theorem
In mathematics, Hall's marriage theorem, proved by , is a theorem with two equivalent formulations: * The combinatorial formulation deals with a collection of finite sets. It gives a necessary and sufficient condition for being able to select a distinct element from each set. * The graph theoretic formulation deals with a bipartite graph. It gives a necessary and sufficient condition for finding a matching that covers at least one side of the graph. Combinatorial formulation Statement Let \mathcal F be a family of finite sets. Here, \mathcal F is itself allowed to be infinite (although the sets in it are not) and to contain the same set multiple times. Let X be the union of all the sets in \mathcal F, the set of elements that belong to at least one of its sets. A transversal for F is a subset of X that can be obtained by choosing a distinct element from each set in \mathcal F. This concept can be formalized by defining a transversal to be the image of an injective function f: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics. Life Ore graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled ''Zur Theorie der algebraischen Körper'', supervised by Thoralf Skolem. Ore also studied at Göttingen University, where he learned Emmy Noether's new approach to abstract algebra. He was also a fellow at the Mittag-Leffler Institute in Sweden, and spent some time at the University of Paris. In 1925, he was appointed research assistant at the University of Oslo. Yale University’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928, then to full professor in 1929. In 1931, he became a Sterling Prof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Set (graph Theory)
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 S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening. A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph G. This size is called the independence number of ''G'' and is usually denoted by \alpha(G). The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. As such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
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 U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.87. Notable publications * The 1972 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermodular Function
In mathematics, a function :f\colon \mathbb^k \to \mathbb is supermodular if : f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y) for all x, y \isin \mathbb^, where x \uparrow y denotes the componentwise maximum and x \downarrow y the componentwise minimum of x and y. If −''f'' is supermodular then ''f'' is called submodular, and if the inequality is changed to an equality the function is modular. If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition : \frac \geq 0 \mbox i \neq j. Supermodularity in economics and game theory The concept of supermodularity is used in the social sciences to analyze how one Agent (economics), agent's decision affects the incentives of others. Consider a symmetric game with a smooth payoff function \,f defined over actions \,z_i of two or more players i \in . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: z_i \in [a,b]. In this context, sup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definition Given a graph ''G'' = (''V'', ''E''), a fractional matching in ''G'' is a function that assigns, to each edge ''e'' in ''E'', a fraction ''f''(''e'') in , 1 such that for every vertex ''v'' in ''V'', the sum of fractions of edges adjacent to ''v'' is at most 1: \forall v\in V: \sum_f(e)\leq 1 A matching in the traditional sense is a special case of a fractional matching, in which the fraction of every edge is either 0 or 1: ''f''(''e'') = 1 if ''e'' is in the matching, and ''f''(''e'') = 0 if it is not. For this reason, in the context of fractional matchings, usual matchings are sometimes called ''integral matchings''. The size of an integral matching is the number of edges in the matching, and the matching number \nu(G) of a graph ''G'' is the largest size of a matching in ''G''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular Set Function
In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathematics related to work by Hungarian mathematicians. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.39, and its 2015 impact factor was 0.469. The editor-in-chief is Imre Bárány, honorary editor is Ákos Császár, the editors are the mathematician members of the Hungarian Academy of Sciences. Abstracting and indexing According to the ''Journal Citation Reports'', the journal had a 2020 impact factor of 0.623. This journal is indexed by the following services: * Science Citation Index * Journal Citation Reports/Science Edition * Scopus * Mathematical Reviews * Zentralblatt Math zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Gallai and Jack Edmonds independently discovered it and proved its key properties. The Gallai–Edmonds decomposition of a graph can be found using the blossom algorithm. Properties Given a graph G, its Gallai–Edmonds decomposition consists of three disjoint sets of vertices, A(G), C(G), and D(G), whose union is V(G): the set of all vertices of G. First, the vertices of G are divided into ''essential vertices'' (vertices which are covered by every maximum matching in G) and ''inessential vertices'' (vertices which are left uncovered by at least one maximum matching in G). The set D(G) is defined to contain all the inessential vertices. Essential vertices are split into A(G) and C(G): the set A(G) is defined to contain all essential vertices adjacent to at least one vertex o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
