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Hereditary Property
In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of ''subobject'' depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. In topology In topology, a topological property is said to be ''hereditary'' if whenever a topological space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called ''weakly hereditary'' or ''closed-hereditary''. For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. Connectivity is not weakly hereditary. If ''P'' is a property of a topological space ''X'' and every subspace also has property ''P'', then ''X'' is said to be "hereditarily ''P''". In combinatorics and graph theory The notion of hereditary properties occurs throughout co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a '' topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid Russell's paradox (see ). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the phrase "ultimate class" is often used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Problem Solving
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices. Solutions require sufficient resources and knowledge to attain the goal. Professionals such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planning
Planning is the process of thinking regarding the activities required to achieve a desired goal. Planning is based on foresight, the fundamental capacity for mental time travel. The evolution of forethought, the capacity to think ahead, is considered to have been a prime mover in human evolution. Planning is a fundamental property of intelligent behavior. It involves the use of logic and imagination to visualise not only a desired end result, but the steps necessary to achieve that result. An important aspect of planning is its relationship to forecasting. Forecasting aims to predict what the future will look like, while planning imagines what the future could look like. Planning according to established principles is a core part of many professional occupations, particularly in fields such as management and business. Once a plan has been developed it is possible to measure and assess progress, efficiency and effectiveness. As circumstances change, plans may need to be modi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aanderaa–Karp–Rosenberg Conjecture
In theoretical computer science, the Aanderaa–Karp–Rosenberg conjecture (also known as the Aanderaa–Rosenberg conjecture or the evasiveness conjecture) is a group of related conjectures about the number of questions of the form "Is there an edge between vertex u and vertex v?" that have to be answered to determine whether or not an undirected graph has a particular property such as planarity or bipartiteness. They are named after Stål Aanderaa, Richard M. Karp, and Arnold L. Rosenberg. According to the conjecture, for a wide class of properties, no algorithm can guarantee that it will be able to skip any questions: any algorithm for determining whether the graph has the property, no matter how clever, might need to examine every pair of vertices before it can give its answer. A property satisfying this conjecture is called evasive. More precisely, the Aanderaa–Rosenberg conjecture states that any deterministic algorithm must test at least a constant fraction of all p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Graph Theory
The ''Journal of Graph Theory'' is a peer-reviewed mathematics journal specializing in graph theory and related areas, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. It is published by John Wiley & Sons. The journal was established in 1977 by Frank Harary.Frank Harary a biographical sketch at the ACM site The are [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forbidden Minor
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph and the complete bipartite graph . For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which case it does not belong to the planar graphs). Definition More generally, a forbidden gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robertson–Seymour Theorem
In graph theory, the Robertson–Seymour theorem (also called the graph minor theorem) states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering. Equivalently, every family of graphs that is closed under minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph ''K''5 or the complete bipartite graph ''K''3,3 as minors. The Robertson–Seymour theorem is named after mathematicians Neil Robertson and Paul D. Seymour, who proved it in a series of twenty papers spanning over 500 pages from 1983 to 2004. Before its proof, the statement of the theorem was known as Wagner's conjecture after the German mathematician Klaus Wagner, although Wagner said he never conjectured it. A weaker result for trees is implied by Kruskal's tree theorem, which was conjectured in 1937 by Andrew Vázsonyi and proved in 1960 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Minor
In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph nor the complete bipartite graph ., p. 77; . The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions., theorem 4, p. 78; . For every fixed graph , it is possible to test whether is a minor of an input graph in polynomial time; together with the forbidden minor characterization this implies that every graph property preserved by deletions and contractions may be recognized in polynomial time. Other results and conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have as a minor may be formed by gl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Multipartite Graph
In graph theory, a part of mathematics, a -partite graph is a graph whose vertices are (or can be) partitioned into different independent sets. Equivalently, it is a graph that can be colored with colors, so that no two endpoints of an edge have the same color. When these are the bipartite graphs, and when they are called the tripartite graphs. Bipartite graphs may be recognized in polynomial time but, for any it is NP-complete, given an uncolored graph, to test whether it is -partite. However, in some applications of graph theory, a -partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the graph represent different types of objects. For instance, folksonomies have been modeled mathematically by tripartite graphs in which the three sets of vertices in the graph represent users of a system, resources that the users are tagging, and tags that the users have applied to the resources. A compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
