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Eulerian Matroid
In matroid theory, an Eulerian matroid is a matroid whose elements can be partitioned into a collection of disjoint circuits. Examples In a uniform matroid U^r_n, the circuits are the sets of exactly r+1 elements. Therefore, a uniform matroid is Eulerian if and only if r+1 is a divisor of n. For instance, the n-point lines U^2_n are Eulerian if and only if n is divisible by three. The Fano plane has two kinds of circuits: sets of three collinear points, and sets of four points that do not contain any line. The three-point circuits are the complements of the four-point circuits, so it is possible to partition the seven points of the plane into two circuits, one of each kind. Thus, the Fano plane is also Eulerian. Relation to Eulerian graphs Eulerian matroids were defined by as a generalization of the Eulerian graphs, graphs in which every vertex has even degree. By Veblen's theorem the edges of every such graph may be partitioned into simple cycles, from which it follows that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Theory
In combinatorics, a branch of mathematics, 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 matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of 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 ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawski' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GF(2)
(also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with the notation of -adic integers. is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and , as usual. The elements of may be identified with the two possible values of a bit and to the boolean values ''true'' and ''false''. It follows that is fundamental and ubiquitous in computer science and its logical foundations. Definition GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted and . Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below: If the elements of GF(2) are seen as boolean values, then the addition is th ... [...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-element subset of S is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Oracle
In mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure that can be used to describe the linear dependencies between vectors in a vector space or the spanning trees of a graph, among other applications. The most commonly used oracle of this type is an independence oracle, a subroutine for testing whether a set of matroid elements is independent. Several other types of oracle have also been used; some of them have been shown to be weaker than independence oracles, some stronger, and some equivalent in computational power.; ; . Many algorithms that perform computations on matroids have been designed to take an oracle as input, allowing them to run efficiently without change on many different kinds of matroids, and without additional assumptions about what kind of matroid they are using. For instance, given an independence oracle for any matroid, it is possible to find the minimum weigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial 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] |
Bipartite Matroid
In mathematics, a bipartite matroid is a matroid all of whose circuits have even size. Example A uniform matroid U^r_n is bipartite if and only if r is an odd number, because the circuits in such a matroid have size r+1. Relation to bipartite graphs Bipartite matroids were defined by as a generalization of the bipartite graphs, graphs in which every cycle has even size. A graphic matroid is bipartite if and only if it comes from a bipartite graph.. Duality with Eulerian matroids An Eulerian graph is one in which all vertices have even degree; Eulerian graphs may be disconnected. For planar graphs, the properties of being bipartite and Eulerian are dual: a planar graph is bipartite if and only if its dual graph is Eulerian. As Welsh showed, this duality extends to binary matroids: a binary matroid is bipartite if and only if its dual matroid is an Eulerian matroid, a matroid that can be partitioned into disjoint circuits. For matroids that are not binary, the duality between Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Matroid
In matroid theory, the dual of a matroid M is another matroid M^\ast that has the same elements as M, and in which a set is independent if and only if M has a basis set disjoint from it... Matroid duals go back to the original paper by Hassler Whitney defining matroids.. Reprinted in , pp. 55–79. See in particular section 11, "Dual matroids", pp. 521–524. They generalize to matroids the notions of plane graph duality. Basic properties Duality is an involution: for all M, (M^\ast)^\ast=M. An alternative definition of the dual matroid is that its basis sets are the complements of the basis sets of M. The basis exchange axiom, used to define matroids from their bases, is self-complementary, so the dual of a matroid is necessarily a matroid. The flats of M are complementary to the cyclic sets (unions of circuits) of M^\ast, and vice versa. If r is the rank function of a matroid M on ground set E, then the rank function of the dual matroid is r^\ast(S)=r(E \setminus S)+, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an 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 to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...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 a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Representation
In the mathematical theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations are analogous to group representations; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of linear algebra. A linear matroid is a matroid that has a representation, and an ''F''-linear matroid (for a field ''F'') is a matroid that has a representation using a vector space over ''F''. Matroid representation theory studies the existence of representations and the properties of linear matroids. Definitions A (finite) matroid (E,\mathcal) is defined by a finite set E (the elements of the matroid) and a non-empty family \mathcal of the subsets of E, called the independent sets of the matroid. It is required to satisfy the properties that every subset of an independent set is itself independent, and that if one ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 0) prod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
