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Planarity
Planarity is a puzzle computer game by John Tantalo, based on a concept by Mary Radcliffe at Western Michigan University. The name comes from the concept of planar graphs in graph theory; these are graphs that can be embedded in the Euclidean plane so that no edges intersect. By Fáry's theorem, if a graph is planar, it can be drawn without crossings so that all of its edges are straight line segments. In the planarity game, the player is presented with a circular layout of a planar graph, with all the vertices placed on a single circle and with many crossings. The goal for the player is to eliminate all of the crossings and construct a straight-line embedding of the graph by moving the vertices one by one into better positions. History and versions The game was written in Flash by John Tantalo at Case Western Reserve University. Online popularity and the local notoriety he gained placed Tantalo as one of Cleveland's most interesting people for 2006. It in turn has inspired the ... [...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 if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called ... [...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] |
Circular Layout
In graph drawing, a circular layout is a style of drawing that places the vertices of a graph on a circle, often evenly spaced so that they form the vertices of a regular polygon. Applications Circular layouts are a good fit for communications network topologies such as star or ring networks, and for the cyclic parts of metabolic networks. For graphs with a known Hamiltonian cycle, a circular layout allows the cycle to be depicted as the circle, and in this way circular layouts form the basis of the LCF notation for Hamiltonian cubic graphs. A circular layout may be used on its own for an entire graph drawing, but it also may be used as the layout for smaller clusters of vertices within a larger graph drawing, such as its biconnected components, clusters of genes in a gene interaction graph, or natural subgroups within a social network. If multiple vertex circles are used in this way, other methods such as force-directed graph drawing may be used to arrange the clusters. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossing Number (graph Theory)
In graph theory, the crossing number of a graph is the lowest number of edge crossings of a plane drawing of the graph . For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing. The study of crossing numbers originated in Turán's brick factory problem, in which Pál Turán asked for a factory plan that minimized the number of crossings between tracks connecting brick kilns to storage sites. Mathematically, this problem can be formalized as asking for the crossing number of a complete bipartite graph. The same problem arose independently in sociology at approximately the same time, in connection with the construction of sociograms. Turán's conjectured formula for the crossing numbers of complete bipartite graphs remains unproven, as does an analogous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science (journal)
''Theoretical Computer Science'' (TCS) is a computer science journal published by Elsevier, started in 1975 and covering theoretical computer science. The journal publishes 52 issues a year. It is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports, its 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... is 0.827. References Computer science journals Elsevier academic journals Publications established in 1975 {{comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called . The number of vertices in equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. Terminology There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or ''n''-gon are also often used. The term ''n''-cycle is sometimes used in other settings. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Properties A cycle graph is: * 2-edge colorable, if and only if it has an even number of vertices * 2-regular * 2-v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Graph Algorithms And Applications
The ''Journal of Graph Algorithms and Applications'' is an open access peer-reviewed scientific journal covering the subject of graph algorithms and graph drawing. The journal was established in 1997 and the editor-in-chief is Giuseppe Liotta (University of Perugia). It is abstracted and indexed by Scopus and MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal '' Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links .... MathSciNet, retrieved 2011-03-02. References External links *{{Official website, http://jgaa.info/ ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Eppstein
David Arthur Eppstein (born 1963) is an American computer scientist and mathematician. He is a Distinguished Professor of computer science at the University of California, Irvine. He is known for his work in computational geometry, graph algorithms, and recreational mathematics. In 2011, he was named an ACM Fellow. Biography Born in Windsor, England, in 1963, Eppstein received a B.S. in Mathematics from Stanford University in 1984, and later an M.S. (1985) and Ph.D. (1989) in computer science from Columbia University, after which he took a postdoctoral position at Xerox's Palo Alto Research Center. He joined the UC Irvine faculty in 1990, and was co-chair of the Computer Science Department there from 2002 to 2005. In 2014, he was named a Chancellor's Professor. In October 2017, Eppstein was one of 396 members elected as fellows of the American Association for the Advancement of Science. Eppstein is also an amateur digital photographer. Research interests In compute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorica
''Combinatorica'' is an international journal of mathematics, publishing papers in the fields of combinatorics and computer science. It started in 1981, with László Babai and László Lovász as the editors-in-chief with Paul Erdős as honorary editor-in-chief. The current editors-in-chief are Imre Bárány and József Solymosi. The advisory board consists of Ronald Graham, Gyula O. H. Katona, Miklós Simonovits, Vera Sós, and Endre Szemerédi. It is published by the János Bolyai Mathematical Society and Springer Verlag. The following members of the ''Hungarian School of Combinatorics'' have strongly contributed to the journal as authors, or have served as editors: Miklós Ajtai, László Babai, József Beck, András Frank, Péter Frankl, Zoltán Füredi, András Hajnal, Gyula Katona, László Lovász, László Pyber, Alexander Schrijver, Miklós Simonovits, Vera Sós, Endre Szemerédi, Tamás Szőnyi, Éva Tardos, Gábor Tardos.{{cite web, url=https://www.spr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
