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Kuratowski's Theorem
In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K_5 (the complete graph on five vertices) or of K_ (a complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph). Statement A planar graph is a graph whose vertices can be represented by points in the Euclidean plane, and whose edges can be represented by simple curves in the same plane connecting the points representing their endpoints, such that no two curves intersect except at a common endpoint. Planar graphs are often drawn with straight line segments representing their edges, but by Fáry's theorem this makes no difference to their graph-theoretic characterization. A subdivision of a graph is a graph formed by subdividing its edges into paths of one or mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GP92-Kuratowski
The EMD GP9 is a four-axle diesel-electric locomotive built by General Motors' Electro-Motive Division between 1954 and 1959. The GP9 succeeded the GP7 as the second model of EMD's General Purpose (GP) line, incorporating a new sixteen-cylinder engine which generated . This locomotive type was offered both with and without control cabs; locomotives built without control cabs were called GP9B locomotives. EMD constructed 3,626 GP9s, including 165 GP9Bs. An additional 646 GP9s were built by General Motors Diesel, EMD's Canadian subsidiary, for a total of 4,257 GP9s produced when Canadian production ended in 1963. The GP9 was succeeded by the similar but slightly more powerful GP18. Design and Production EMD designed the GP9 as an improved version of the GP7, with an increase in power from 1,500 hp to 1,750 hp, and a change in prime mover to the latest version of the 567 engine, the 567C. Externally, the GP9 strongly resembled its predecessor. Most were built with high short hoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) are adjacent in ''H''. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as G\simeq H. In the case when the bijection is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the bijection is called an automorphism of ''G''. If a graph is finite, we can prove it to be bijective by showing it is one-one/onto; no need to show both. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due to an unsuccessful eye surgery after a primus stove explosion when he was 14. Despite his blindness he was able to become one of the greatest mathematicians of the 20th century, partially with the help of his mother Tatyana Andreevna who read mathematical books and papers (notably those of Heinz Hopf, J. H. C. Whitehead, and Hassler Whitney) to him. He made major discoveries in a number of fields of mathematics, including optimal control, algebraic topology and differential topology. Work Pontryagin worked on duality theory for homology while still a student. He went on to lay foundations for the abstract theory of the Fourier transform, now called Pontryagin duality. With René Thom, he is regarded as one of the co-founders of cobordism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soviet Union
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national republics; in practice, both its government and its economy were highly centralized until its final years. It was a one-party state governed by the Communist Party of the Soviet Union, with the city of Moscow serving as its capital as well as that of its largest and most populous republic: the Russian SFSR. Other major cities included Leningrad (Russian SFSR), Kiev (Ukrainian SSR), Minsk ( Byelorussian SSR), Tashkent (Uzbek SSR), Alma-Ata (Kazakh SSR), and Novosibirsk (Russian SFSR). It was the largest country in the world, covering over and spanning eleven time zones. The country's roots lay in the October Revolution of 1917, when the Bolsheviks, under the leadership of Vladimir Lenin, overthrew the Russian Provisional Government ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Menger
Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curves and regions; in graph theory, he is credited with Menger's theorem. Outside of mathematics, Menger has substantial contributions to game theory and social sciences. Biography Karl Menger was a student of Hans Hahn and received his PhD from the University of Vienna in 1924. L. E. J. Brouwer invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at Harvard University and the Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame. From 1946 to 1971, he was a professor at Illinois Institute of Technology (IIT) in Chicago. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Althaus Smith
Paul Althaus Smith (May 18, 1900June 13, 1980) was an American mathematician. His name occurs in two significant conjectures in geometric topology: the Smith conjecture, which is now a theorem, and the Hilbert–Smith conjecture, which was proved in dimension 3 in 2013. ''Smith theory'' is a theory about homeomorphisms of finite order of manifolds, particularly spheres. Smith was a student of Solomon Lefschetz at the University of Kansas, moving to Princeton University with Lefschetz in the mid-1920s. He finished his doctorate at Princeton, in 1926. His Ph.D. thesis was published in the ''Annals of Mathematics'' that same year. He also worked with George David Birkhoff, with whom he wrote a 1928 paper in ergodic theory, entitled ''Structure analysis of surface transformations'', which appeared in the ''Journal des Mathématiques''. He subsequently became a professor at Columbia University and at Barnard College. His students at Columbia included Sherman K. Stein and Moses Ric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orrin Frink
Orrin Frink Jr. (31 May 1901 – 4 March 1988). was an American mathematician who introduced Frink ideals in 1954. Frink earned a doctorate from Columbia University in 1926 or 1927 and worked on the faculty of Pennsylvania State University for 41 years, 11 of them as department chair. His time at Penn State was interrupted by service as assistant chief engineer at the Special Projects Laboratory at Wright-Patterson Air Force Base during World War II, and by two Fulbright fellowships to Dublin, Ireland in the 1960s. Aline Huke Frink, his wife, was also a mathematician at Penn State. Their son, also named Orrin Frink, became a professor of Slavic languages at Ohio University and Iowa State University.. Selected publications * * * See also *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 cubic, bridgeless gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lecture Notes In Computer Science
''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post-proceedings, monographs, and Festschrifts. In addition, tutorials, state-of-the-art surveys, and "hot topics" are increasingly being included. The series is indexed by DBLP. See also *''Monographiae Biologicae'', another monograph series published by Springer Science+Business Media *''Lecture Notes in Physics'' *''Lecture Notes in Mathematics'' *''Electronic Workshops in Computing ''Electronic Workshops in Computing'' (eWiC) is a publication series by the British Computer Society. The series provides free online access for conferences and workshops in the area of computing. For example, the EVA London Conference proceeding ...'', published by the British Computer Society References External links * Publications established in 1973 Computer science books Series of non-fiction books Springer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch And Cut
Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten the linear programming relaxations. Note that if cuts are only used to tighten the initial LP relaxation, the algorithm is called branch and cut. Algorithm This description assumes the ILP is a maximization problem. The method solves the linear program without the integer constraint using the regular simplex algorithm. When an optimal solution is obtained, and this solution has a non-integer value for a variable that is supposed to be integer, a cutting plane algorithm may be used to find further linear constraints which are satisfied by all feasible integer points but violated by the current fractional solution. These inequalities may be added to the linear program, such that r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planarity Testing
In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can be drawn in the plane without edge intersections). This is a well-studied problem in computer science for which many practical algorithms have emerged, many taking advantage of novel data structures. Most of these methods operate in O(''n'') time (linear time), where ''n'' is the number of edges (or vertices) in the graph, which is asymptotically optimal. Rather than just being a single Boolean value, the output of a planarity testing algorithm may be a planar graph embedding, if the graph is planar, or an obstacle to planarity such as a Kuratowski subgraph if it is not. Planarity criteria Planarity testing algorithms typically take advantage of theorems in graph theory that characterize the set of planar graphs in terms that are independent of graph drawings. These include *Kuratowski's theorem that a graph is planar if and only i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
