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Trapezoid Graph
In graph theory, trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. They are a class of co-comparability graphs that contain interval graphs and permutation graphs as subclasses. A graph is a trapezoid graph if there exists a set of trapezoids corresponding to the vertices of the graph such that two vertices are joined by an edge if and only if the corresponding trapezoids intersect. Trapezoid graphs were introduced by Dagan, Golumbic, and Pinter in 1988. There exists (n\log n) algorithms for chromatic number, weighted independent set, clique cover, and maximum weighted clique. Definitions and characterizations Given a channel, a pair of two horizontal lines, a trapezoid between these lines is defined by two points on the top and two points on the bottom line. A graph is a trapezoid graph if there exists a set of trapezoids corresponding to the vertices of the graph such that two vertices are joined by an edge if and only if the corresponding ... [...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] |
Intersection Graph
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them. Formal definition Formally, an intersection graph is an undirected graph formed from a family of sets : S_i, \,\,\, i = 0, 1, 2, \dots by creating one vertex for each set , and connecting two vertices and by an edge whenever the corresponding two sets have a nonempty intersection, that is, : E(G) = \. All graphs are intersection graphs Any undirected graph may be represented as an intersection graph. For each vertex of , form a set consisting of the edges incident to ; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Therefore, is the intersection graph of the sets . provide a construction that is more ef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadrilateral in Euclidean geometry. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are called the ''legs'' (or the ''lateral sides'') if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases). A ''scalene trapezoid'' is a trapezoid with no sides of equal measure, in contrast with the #Special cases, special cases below. Etymology and ''trapezium'' versus ''trapezoid'' Ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια (''trapezia'' literally "a table", itself fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Graph
In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs. They can be recognized in linear time, and an optimal graph coloring or maximum clique in these graphs can be found in linear time. The interval graphs include all proper interval graphs, graphs defined in the same way from a set of unit intervals. These graphs have been used to model food webs, and to study scheduling problems in which one must select a subset of tasks to be performed at non-overlapping times. Other applications include assembling contiguous subsequences in DNA mapping, and temporal reasoning. Definition An interval graph is an undirected graph formed from a family of intervals :S_i,\quad i=0,1,2,\dots by creating one vertex for each interval , and connecting two ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Graph
In the mathematical field of graph theory, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition. Definition and characterization If \rho = (\sigma_1,\sigma_2,...,\sigma_n) is any permutation of the numbers from 1 to n, then one may define a permutation graph from \sigma in which there are n vertices v_1, v_2, ..., v_n, and in which there is an edge v_i v_j for any two indices i and j for which i\sigma_j. That is, two indices i and j determine an edge in the permutation graph exactly when they determine an inversion in the permutatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ido Dagan
Ido () is a constructed language derived from Reformed Esperanto, and similarly designed with the goal of being a universal second language for people of diverse backgrounds. To function as an effective ''international auxiliary language'', Ido was specifically designed to be grammatically, orthographically, and lexicographically regular (and, above all, easy to learn and use). It is the most successful of the many Esperanto derivatives, called ''Esperantidoj''. Ido was created in 1907 out of a desire to reform perceived flaws in Esperanto, a language that had been created 20 years earlier to facilitate international communication. The name of the language traces its origin to the Esperanto word ', meaning "offspring", since the language is a "descendant" of Esperanto. After its inception, Ido gained support from some in the Esperanto community. A setback occurred with the sudden death in 1914 of one of its most influential proponents, Louis Couturat. In 1928, leader Otto Jes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Charles Golumbic
Martin Charles Golumbic (born 1948) is a mathematician and computer scientist known for his research on perfect graphs, graph sandwich problems, compiler optimization, and spatial-temporal reasoning. He is a professor emeritus of computer science at the University of Haifa, and was the founder of the journal ''Annals of Mathematics and Artificial Intelligence''. Education and career Golumbic majored in mathematics at Pennsylvania State University, graduating in 1970 with bachelor's and master's degrees. He completed his Ph.D. at Columbia University in 1975, with the dissertation ''Comparability Graphs and a New Matroid'' supervised by Samuel Eilenberg. He became an assistant professor in the Courant Institute of Mathematical Sciences of New York University from 1975 until 1980, when he moved to Bell Laboratories. From 1983 to 1992 he worked for IBM Research in Israel, and from 1992 to 2000 he was a professor of mathematics and computer science at Bar-Ilan University. He moved to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ron Pinter
Ron Yair Pinter ( he, רון יאיר פינטר) is an Israeli computer scientist specializing in computational systems biology, integrated circuit layout and compiler optimization. He is Professor of Computer Science and the Rappaport Medical School at the Technion in Haifa, Israel. He was a founding member of the Israeli Society for Bioinformatics and Computational Biology, . In the past, he has been a Program Manager at the IBM Haifa Research Laboratory and a member of the IBM Academy of Technology, and Vice President for Research and Development at Compugen. He is an author and co-author of more than 90 books and peer-reviewed articles, all of which were cited more than 2500 times. His contributions include defining (with Ido Dagan and Martin Golumbic) the notion of trapezoid graphs, and pioneering analysis of biological networks. Biography Pinter was born in Haifa, Israel in 1953 to parents of German Jewish descent both refugees from Nazi Germany in the 1930s. After fin ... [...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] |
Very-large-scale Integration
Very large-scale integration (VLSI) is the process of creating an integrated circuit (IC) by combining millions or billions of MOS transistors onto a single chip. VLSI began in the 1970s when MOS integrated circuit (Metal Oxide Semiconductor) chips were developed and then widely adopted, enabling complex semiconductor and telecommunication technologies. The microprocessor and memory chips are VLSI devices. Before the introduction of VLSI technology, most ICs had a limited set of functions they could perform. An electronic circuit might consist of a CPU, ROM, RAM and other glue logic. VLSI enables IC designers to add all of these into one chip. History Background The history of the transistor dates to the 1920s when several inventors attempted devices that were intended to control current in solid-state diodes and convert them into triodes. Success came after World War II, when the use of silicon and germanium crystals as radar detectors led to improvements in fabrication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Graph
In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph (clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is perfect if and only if for all S\subseteq V we have \chi(G =\omega(G . The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time. In addition, several important min-max theorems in combinatorics, such as Dilworth's theorem, can be expressed in terms of the perfection of certain associated graphs. A graph G is 1-perfect if and only if \chi(G)=\omega(G). Then, G is perfect if and only if every induced subgraph of G is 1-perfect. Properties * By the perfect graph theorem, a graph G is perfect if and on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
