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Erdős–Menger Conjecture
In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs. Edge connectivity The edge-connectivity version of Menger's theorem is as follows: :Let ''G'' be a finite undirected graph and ''x'' and ''y'' two distinct vertices. Then the size of the minimum edge cut for ''x'' and ''y'' (the minimum number of edges whose removal disconnects ''x'' and ''y'') is equal to the maximum number of pairwise edge-disjoint paths from ''x'' to ''y''. The implication for the graph ''G'' is the following version: :A graph is ''k''-edge-connected (it remains connected after removing fe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max-flow Min-cut Theorem
In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the ''source'' to the ''sink'' is equal to the total weight of the edges in a minimum cut, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink. For example, imagine a network of pipes carrying water from a reservoir (the source) to a city (the sink). Each pipe has a capacity representing the maximum amount of water that can flow through it per unit of time. The max-flow min-cut theorem tells us that the maximum amount of water that can reach the city is limited by the smallest total capacity of any set of pipes that, if cut, would completely isolate the reservoir from the city. This smallest total capacity is the min-cut. So, if there's a bottleneck in the pipe network, represented by a small min-cut, that bottleneck will determine the overall maximum flow of water to the city. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Connectivity
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two vertices and are called connected if contains a path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length (that is, they are the endpoints of a single edge), the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. An undirected graph is therefore disconnected if there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Separator
In graph theory, a vertex subset is a vertex separator (or vertex cut, separating set) for nonadjacent Vertex (graph theory), vertices and if the Graph partition, removal of from the Graph (discrete mathematics), graph separates and into distinct connected component (graph theory), connected components. Examples Consider a grid graph with rows and columns; the total number of vertices is . For instance, in the illustration, , , and . If is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if is odd, there is a single central column, and otherwise there are two columns equally close to the center. Choosing to be any of these central rows or columns, and removing from the graph, partitions the graph into two smaller connected subgraphs and , each of which has at most vertices. If (as in the illustration), then choosing a central column will give a separator with r \leq \sqrt vertices, and similarly if the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-edge-connected Graph
In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If the subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is said to be ''k''-edge-connected. The edge connectivity of is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a minimum cut in ''G''. The edge connectivity version of Menger's theorem provides an alternative and equivalent character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected Graph (discrete mathematics), graph is said to be -vertex-connected (or -connected) if it has more than Vertex (graph theory), vertices and remains Connectivity (graph theory), connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. In complete graphs, there is no subset whose removal would disconnect the graph. Some sources modify the definition of connectivity to handle this case, by defining it as the size of the smallest subset of vertices whose deletion results in either a disconnected graph or a single vertex. For this variation, the connectivity of a complete graph K_n is n-1. An equivalent definition is that a graph with at least two vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gammoid
In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of Vertices (graph theory), vertices that can be reached by vertex-disjoint Path (graph theory), paths in a directed graph. The concept of a gammoid was introduced and shown to be a matroid by , based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths. Gammoids were given their name by . and studied in more detail by .. Definition Let G be a directed graph, S be a set of starting vertices, and T be a set of destination vertices (not necessarily disjoint from S). The gammoid \Gamma derived from this data has T as its set of elements. A subset I of T is independent in \Gamma if there exists a set of vertex-disjoint paths whose starting points all belong to S and whose ending points are exactly I.. A strict gammoid is a gammoid in which the set T of destination vertices consists of every vertex in G. Thus, a ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; ''Time'' magazine called him "The Oddball's Oddba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eli Berger
Eli most commonly refers to: * Eli (name), a given name, nickname and surname * Eli (biblical figure) Eli or ELI may also refer to: Film * ''Eli'' (2015 film), a Tamil film * ''Eli'' (2019 film), an American horror film Music * ''Eli'' (Jan Akkerman album) (1976) * ''Eli'' (Supernaut album) (2006) Places * Alni, Ardabil Province, Iran, also known as Elī * Eli, Mateh Binyamin, an Israeli settlement in the West Bank * Éile or Éli, a medieval kingdom in Ireland * Eli, Kentucky, United States * Eli, Nebraska, United States * Eli, West Virginia, United States Other uses * ''Eli'' (opera), an opera by Walter Steffens * ELI (programming language) * Earth Learning Idea * English language institute * Environmental Law Institute, an American environmental law policy organization * European Law Institute * European Legislation Identifier * Extreme Light Infrastructure, a high energy laser research facility of the European Union * Eli, someone from Yale University, after Elihu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ron Aharoni
Ron Aharoni (; born 1952) is an Israeli mathematician, working in finite and infinite combinatorics. Aharoni is a professor at the Technion – Israel Institute of Technology, where he received his Ph.D. in mathematics in 1979. With Nash-Williams and Shelah he generalized Hall's marriage theorem by obtaining the right transfinite conditions for infinite bipartite graphs. He subsequently proved the appropriate versions of the Kőnig theorem and the Menger theorem for infinite graph This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B ...s (the latter with Eli Berger). Aharoni is the author of several nonspecialist books; the most successful is '' Arithmetic for Parents'', a book helping parents and elementary school teachers in teaching basic mathematics. He also wrote a book on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
End (graph Theory)
In the mathematics of infinite graphs, an end of an undirected graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite path (graph theory), paths, as Haven (graph theory), havens describing strategies for pursuit–evasion games on the graph, or (in the case of locally finite graphs) as end (topology), topological ends of topological spaces associated with the graph. Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end. Definition and characterization Ends of graphs were defined by in terms of equivalence classes of infinite paths. A in an infinite graph is a semi-infinite simple path (graph theory), simple path; that is, it is an infinite sequence of vertices v_0,v_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), 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 cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, 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 Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
