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Circuit Rank
In graph theory, a branch of mathematics, the cyclomatic number, circuit rank, cycle rank, or nullity of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. Formula The cyclomatic number of a graph equals the number of independent cycles in the graph, the size of a cycle basis. Unlike the corresponding feedback arc set problem for directed graphs, the cyclomatic number is easily computed using the formula: r = e - v + c, where is the number of edges in the given graph, is the number of vertices, and is the number of connected components. . It is possible to construct a minimum-size set of edges that breaks all cycles efficiently, either using a greedy algorithm or by complementing a spanning forest. The cyclomatic number can be explained in terms of algebraic graph theory as the dimension of the cycle space of a graph, in terms of matroid theory as the corank of a graphic m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Theory
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 Axiomatic system, 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 Cryptomorphism, 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 ''gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Rank
In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset ''S'' of elements of the matroid is, similarly, the maximum size of an independent subset of ''S'', and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized. Matroid rank functions form an important subclass of the submodular set functions. The rank functions of matroids defined from certain other types of mathematical object such as undirected graphs, matrices, and field extensions are important within the study of those objects. Examples In all examples, ''E'' is the base set of the matroid, and ''B'' is some subset of ''E''. * Let ''M'' be the free matroid, where the independent sets are all subsets of ''E''. Then the rank function of ''M'' is simply: ''r''(''B'') = , ''B'', . * Let ''M'' be a uniform mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Handshaking Lemma
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. The handshaking lemma is a consequence of the degree sum formula, also sometimes called the handshaking lemma, according to which the sum of the Degree (graph theory), degrees (the numbers of times each vertex is touched) equals twice the number of edges in the graph. Both results were proven by in his famous paper on the Seven Bridges of Königsberg that began the study of graph theory. Beyond the Seven Bridges of Königsberg Problem, which subsequently formalized Euler tour, Eulerian Tours, other applications of the degree sum formula include proofs of certain combinatorial structures. For example, in the proofs of Sperner's lemma and the mountain climbing problem the geometric pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Graph
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4-cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure. Levi graphs of configurations are biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.. Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergraph
In mathematics, a hypergraph is a generalization of a Graph (discrete mathematics), graph in which an graph theory, edge can join any number of vertex (graph theory), vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair (X,E), where X is a set of elements called ''nodes'', ''vertices'', ''points'', or ''elements'' and E is a set of pairs of subsets of X. Each of these pairs (D,C)\in E is called an ''edge'' or ''hyperedge''; the vertex subset D is known as its ''tail'' or ''domain'', and C as its ''head'' or ''codomain''. The order of a hypergraph (X,E) is the number of vertices in X. The size of the hypergraph is the number of edges in E. The order of an edge e=(D,C) in a directed hypergraph is , e, = (, D, ,, C, ): that is, the number of vertices in its tail followed by the number of vertices in its head. The definition above generalizes from a directed graph to a directed hypergraph by defining the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gustav Kirchhoff
Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German chemist, mathematician, physicist, and spectroscopist who contributed to the fundamental understanding of electrical circuits, spectroscopy and the emission of black-body radiation by heated objects. He also coined the term ''black body'' in 1860. Several different sets of concepts are named "Kirchhoff's laws" after him, which include Kirchhoff's circuit laws, Kirchhoff's law of thermal radiation, and Kirchhoff's law of thermochemistry. The Bunsen–Kirchhoff Award for spectroscopy is named after Kirchhoff and his colleague, Robert Bunsen. Life and work Gustav Kirchhoff was born on 12 March 1824 in Königsberg, Prussia, the son of Friedrich Kirchhoff, a lawyer, and Johanna Henriette Wittke. His family were Lutheranism, Lutherans in the Evangelical Church of Prussia. He graduated from the Albertus University of Königsberg in 1847 where he attended the mathematico-physical seminar directed by Carl Gusta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclomatic Complexity
Cyclomatic complexity is a software metric used to indicate the complexity of a program. It is a quantitative measure of the number of linearly independent paths through a program's source code. It was developed by Thomas J. McCabe, Sr. in 1976. Cyclomatic complexity is computed using the control-flow graph of the program. The nodes of the graph correspond to indivisible groups of commands of a program, and a directed edge connects two nodes if the second command might be executed immediately after the first command. Cyclomatic complexity may also be applied to individual functions, modules, methods, or classes within a program. One testing strategy, called basis path testing by McCabe who first proposed it, is to test each linearly independent path through the program. In this case, the number of test cases will equal the cyclomatic complexity of the program. Description Definition There are multiple ways to define cyclomatic complexity of a section of source code. One com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Software Metric
In software engineering and development, a software metric is a standard of measure of a degree to which a software system or process possesses some property. Even if a metric is not a measurement (metrics are functions, while measurements are the numbers obtained by the application of metrics), often the two terms are used as synonyms. Since quantitative measurements are essential in all sciences, there is a continuous effort by computer science practitioners and theoreticians to bring similar approaches to software development. The goal is obtaining objective, reproducible and quantifiable measurements, which may have numerous valuable applications in schedule and budget planning, cost estimation, quality assurance, testing, software debugging, software performance optimization, and optimal personnel task assignments. Common software measurements Common software measurements include: * ABC Software Metric * Balanced scorecard * Bugs per line of code * Code coverage * Coh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameterized Complexity
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to ''multiple'' parameters of the input or output. The complexity of a problem is then measured as a function of those parameters. This allows the classification of NP-hard problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input. This appears to have been first demonstrated in . The first systematic work on parameterized complexity was done by . Under the assumption that P ≠ NP, there exist many natural problems that require super-polynomial running time when complexity is measured in terms of the input size only but that are computable in a time that is polynomial in the input size and exponential or worse in a parameter . Hence, if is fixed at a small value and the growth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ear Decomposition
In graph theory, an ear of an undirected graph ''G'' is a path (graph theory), path ''P'' where the two endpoints of the path may coincide, but where otherwise no repetition of edges or vertices is allowed, so every internal vertex of ''P'' has degree (graph theory), degree two in ''G''. An ear decomposition of ''G'' is a Partition of a set, partition of its set of edges into a sequence of ears, such that the one or two endpoints of each ear belong to earlier ears in the sequence and such that the internal vertices of each ear do not belong to any earlier ear. Often, the first ear in the sequence is taken to be a cycle. An open ear decomposition or a proper ear decomposition is an ear decomposition in which the two endpoints of each ear after the first are distinct from each other. Ear decompositions may be used to characterize several important graph classes, and as part of efficient graph algorithms. They may also be generalized from graphs to matroids. Characterizing graph cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
