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Circular-arc Graph
In graph theory, a circular-arc graph is the intersection graph of a set of arcs on the circle. It has one vertex for each arc in the set, and an edge between every pair of vertices corresponding to arcs that intersect. Formally, let :I_1, I_2, \ldots, I_n \subset C_1 be a set of arcs. Then the corresponding circular-arc graph is ''G'' = (''V'', ''E'') where : V = \ and : \ \in E \iff I_\alpha \cap I_\beta \neq \varnothing. A family of arcs that corresponds to G is called an ''arc model''. Recognition demonstrated the first polynomial recognition algorithm for circular-arc graphs, which runs in (n^3) time. gave the first linear ((n+m)) time recognition algorithm, where m is the number of edges. More recently, Kaplan and Nussbaum developed a simpler linear time recognition algorithm. Relation to other graph classes Circular-arc graphs are a natural generalization of interval graphs. If a circular-arc graph ''G'' has an arc model that leaves some point of the cir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular-arc Graph
In graph theory, a circular-arc graph is the intersection graph of a set of arcs on the circle. It has one vertex for each arc in the set, and an edge between every pair of vertices corresponding to arcs that intersect. Formally, let :I_1, I_2, \ldots, I_n \subset C_1 be a set of arcs. Then the corresponding circular-arc graph is ''G'' = (''V'', ''E'') where : V = \ and : \ \in E \iff I_\alpha \cap I_\beta \neq \varnothing. A family of arcs that corresponds to G is called an ''arc model''. Recognition demonstrated the first polynomial recognition algorithm for circular-arc graphs, which runs in (n^3) time. gave the first linear ((n+m)) time recognition algorithm, where m is the number of edges. More recently, Kaplan and Nussbaum developed a simpler linear time recognition algorithm. Relation to other graph classes Circular-arc graphs are a natural generalization of interval graphs. If a circular-arc graph ''G'' has an arc model that leaves some point of the cir ... [...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] |
Arc (geometry)
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is €¦the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which €¦will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (graph Theory)
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 C D E F G H I K L M N O ... [...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] |
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] |
Claw-free Graph
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph ''K''1,3 (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, and the characterization of claw-free perfect graphs., p. 88. They are the subject of hundreds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helly Family
In combinatorics, a Helly family of order is a family of sets in which every minimal ''subfamily with an empty intersection'' has or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non-empty total intersection.. The -Helly property is the property of being a Helly family of order .. See in particular Section 2.5, "Helly Property"pp. 393–394 The number is frequently omitted from these names in the case that . Thus, a set-family has the Helly property if, for every sets s_1,\ldots,s_n in the family, if \forall i,j\in s_i \cap s_j \neq\emptyset , then s_1 \cap \cdots \cap s_n \neq\emptyset . These concepts are named after Eduard Helly (1884–1943); Helly's theorem on convex sets, which gave rise to this notion, states that convex sets in Euclidean space of dimension are a Helly family of order . Examples * In the family of all subsets of the set , the subfamily has an empty intersection, but removing any s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resource Allocation
In economics, resource allocation is the assignment of available resources to various uses. In the context of an entire economy, resources can be allocated by various means, such as markets, or planning. In project management, resource allocation or resource management is the scheduling of activities and the resources required by those activities while taking into consideration both the resource availability and the project time. Economics In economics, the field of public finance deals with three broad areas: macroeconomic stabilization, the distribution of income and wealth, and the allocation of resources. Much of the study of the allocation of resources is devoted to finding the conditions under which particular mechanisms of resource allocation lead to Pareto efficient outcomes, in which no party's situation can be improved without hurting that of another party. Strategic planning In strategic planning, resource allocation is a plan for using available resources, for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operations Research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decision-making. It is considered to be a subfield of mathematical sciences. The term management science is occasionally used as a synonym. Employing techniques from other mathematical sciences, such as modeling, statistics, and optimization, operations research arrives at optimal or near-optimal solutions to decision-making problems. Because of its emphasis on practical applications, operations research has overlap with many other disciplines, notably industrial engineering. Operations research is often concerned with determining the extreme values of some real-world objective: the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost). Originating in military efforts before World War II, its techniques have grown to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
