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Uniform Matroid
In mathematics, a uniform matroid is a matroid in which the ''independent sets'' are exactly the sets containing at most ''r'' elements, for some fixed integer ''r''. An alternative definition is that every permutation of the elements is a symmetry. Definition The uniform matroid U^r_n is defined over a set of n elements. A subset of the elements is independent if and only if it contains at most r elements. A subset is a basis if it has exactly r elements, and it is a circuit if it has exactly r+1 elements. The rank of a subset S is \min(, S, ,r) and the rank of the matroid is r. A matroid of rank r is uniform if and only if all of its circuits have exactly r+1 elements. The matroid U^2_n is called the n-point line. Duality and minors The dual matroid of the uniform matroid U^r_n is another uniform matroid U^_n. A uniform matroid is self-dual if and only if r=n/2. Every minor of a uniform matroid is uniform. Restricting a uniform matroid U^r_n by one element (as long as r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphic Matroid Of C4
Graphics () are visual perception, visual Image scanner, images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustration, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of the data, as in design and manufacture, in typesetting and the graphic arts, and in educational and recreational software. Images that are generated by a computer are called Computer graphics (computer science), computer graphics. Examples are photographs, drawings, line art, graph of a function, mathematical graphs, line chart, line graphs, charts, diagrams, typography, numbers, symbols, geometric designs, maps, engineering drawings, or other images. Graphics often combine character (computer), text, illustration, and color. Graphic design may consist of the deliberate selection, creation, or arrangement of typography alone, as in a brochure, flyer, poster, web site, or book without any other element. The objective can be clari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selection Algorithm
In computer science, a selection algorithm is an algorithm for finding the kth smallest value in a collection of ordered values, such as numbers. The value that it finds is called the order statistic. Selection includes as special cases the problems of finding the minimum, median, and maximum element in the collection. Selection algorithms include quickselect, and the median of medians algorithm. When applied to a collection of n values, these algorithms take linear time, O(n) as expressed using big O notation. For data that is already structured, faster algorithms may be possible; as an extreme case, selection in an already-sorted array takes Problem statement An algorithm for the selection problem takes as input a collection of values, and a It outputs the smallest of these values, or, in some versions of the problem, a collection of the k smallest values. For this to be well-defined, it should be possible to sort the values into an order from smallest to largest; ... [...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. If n = 1, it is an isolated loop. 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 n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Graph
In the mathematics, mathematical discipline of graph theory, the dual graph of a planar graph is a graph that has a vertex (graph theory), vertex for each face (graph theory), face of . The dual graph has an edge (graph theory), edge for each pair of faces in that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge of has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of . The definition of the dual depends on the choice of embedding of the graph , so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph Duality (mathematics), duality to be recognized was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dipole Graph
In graph theory, a dipole graph, dipole, bond graph, or linkage, is a multigraph consisting of two vertex (graph theory), vertices connected with a number of Multiple edges, parallel edges. A dipole graph containing edges is called the dipole graph, and is denoted by . The dipole graph is dual graph, dual to the cycle graph . The Hexagonal lattice, honeycomb as an abstract graph is the maximal abelian covering graph of the dipole graph , while the Diamond cubic, diamond crystal as an abstract graph is the maximal abelian covering graph of . Similarly to the Platonic graphs, the dipole graphs form the skeletons of the hosohedron, hosohedra. Their duals, the cycle graphs, form the skeletons of the dihedron, dihedra. References * * Jonathan L. Gross and Jay Yellen, 2006. ''Graph Theory and Its Applications, 2nd Ed.'', p. 17. Chapman & Hall/CRC. * Toshikazu Sunada, Sunada T., ''Topological Crystallography, With a View Towards Discrete Geometric Analysis'', Springer, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphic Matroid
In the mathematical theory of Matroid theory, matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the tree (graph theory), forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs. Definition A matroid may be defined as a family of finite sets (called the "independent sets" of the matroid) that is closed under subsets and that satisfies the "exchange property": if sets A and B are both independent, and A is larger than B, then there is an element x\in A\setminus B such that B\cup\ remains independent. If G is an undirected graph, and F is the family of sets of edges that form forests in G, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gammoid
In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of Vertices (graph theory), vertices that can be reached by vertex-disjoint Path (graph theory), paths in a directed graph. The concept of a gammoid was introduced and shown to be a matroid by , based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths. Gammoids were given their name by . and studied in more detail by .. Definition Let G be a directed graph, S be a set of starting vertices, and T be a set of destination vertices (not necessarily disjoint from S). The gammoid \Gamma derived from this data has T as its set of elements. A subset I of T is independent in \Gamma if there exists a set of vertex-disjoint paths whose starting points all belong to S and whose ending points are exactly I.. A strict gammoid is a gammoid in which the set T of destination vertices consists of every vertex in G. Thus, a ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transversal Matroid
In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or ''flats''. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terms used in both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory, and coding theory. Definition There are many equivalent ways to define a (finite) matroid. Independent sets In terms of independence, a finite matroid M is a pair (E, \mathcal), where E is a finite set (called the ''ground set'') and \mathcal is a fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paving Matroid
In the mathematical theory of matroids, a paving matroid is a matroid in which every circuit has size at least as large as the matroid's rank. In a matroid of rank r every circuit has size at most r+1, so it is equivalent to define paving matroids as the matroids in which the size of every circuit belongs to the set \.. It has been conjectured that almost all matroids are paving matroids. Examples Every simple matroid of rank three is a paving matroid; for instance this is true of the Fano matroid. The Vámos matroid provides another example, of rank four. Uniform matroids of rank r have the property that every circuit is of length exactly r+1 and hence are all paving matroids; the converse does not hold, for example, the cycle matroid of the complete graph K_4 is paving but not uniform. A Steiner system S(t,k,v) is a pair (S,\mathcal) where S is a finite set of size v and \mathcal is a family of k-element subsets of S with the property that every t distinct elements of S are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Matroid
In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a ''capacity constraint'' - a maximum number of allowed elements from this category. The independent sets of a partition matroid are exactly the sets in which, for each category, the number of elements from this category is at most the category capacity. Formal definition Let C_i be a collection of disjoint sets ("categories"). Let d_i be integers with 0\le d_i\le , C_i, ("capacities"). Define a subset I\subseteq \bigcup_i C_i to be "independent" when, for every index i, , I\cap C_i, \le d_i. The sets satisfying this condition form the independent sets of a matroid, called a partition matroid. The sets C_i are called the categories or the blocks of the partition matroid. A basis of the partition matroid is a set whose intersection with every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
