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Binary Matroid
In matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2).. That is, up to isomorphism, they are the matroids whose elements are the columns of a (0,1)-matrix and whose sets of elements are independent if and only if the corresponding columns are linearly independent in GF(2). Alternative characterizations A matroid M is binary if and only if *It is the matroid defined from a symmetric (0,1)-matrix. *For every set \mathcal of circuits of the matroid, the symmetric difference of the circuits in \mathcal can be represented as a disjoint union of circuits., Theorem 10.1.3, p. 162. *For every pair of circuits of the matroid, their symmetric difference contains another circuit. *For every pair C,D where C is a circuit of M and D is a circuit of the dual matroid of M, , C\cap D, is an even number.. *For every pair B,C where B is a basis of M and C is a circuit of M, C is the symmetric difference of the fundamental circuits induced in B by the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
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 Brylawsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Matroid
In mathematics, a regular matroid is a matroid that can be represented over all fields. Definition A matroid is defined to be a family of subsets of a finite set, satisfying certain axioms. The sets in the family are called "independent sets". One of the ways of constructing a matroid is to select a finite set of vectors in a vector space, and to define a subset of the vectors to be independent in the matroid when it is linearly independent in the vector space. Every family of sets constructed in this way is a matroid, but not every matroid can be constructed in this way, and the vector spaces over different fields lead to different sets of matroids that can be constructed from them. A matroid M is regular when, for every field F, M can be represented by a system of vectors over F.. Properties If a matroid is regular, so is its dual matroid, and so is every one of its minors. Every direct sum of regular matroids remains regular. Every graphic matroid (and every co-graphic matro ... [...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] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graphs
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] |
Eulerian Graph
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this: :Given the graph in the image, is it possible to construct a path (or a cycle; i.e., a path starting and ending on the same vertex) that visits each edge exactly once? Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: :A connected g ... [...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] |
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] |
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] |
Cactus Graph
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cactus) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle. Properties Cacti are outerplanar graphs. Every pseudotree is a cactus. A nontrivial graph is a cactus if and only if every block is either a simple cycle or a single edge. The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph ''K''4. Triangular cactus A triangular cactus is a special type of cactus graph such that each cycle has length three and each edge belongs to a cycle. For instance, the friendship graphs, graphs formed fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Plane
In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is . Here stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one). The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study. Homogeneous coordinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphic Matroid
In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the 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 F is clearly closed under subsets (re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
