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Cyclic Graph
In mathematics, a cyclic graph may mean a graph that contains a cycle, or a graph that is a cycle, with varying definitions of cycles. See: *Cycle (graph theory), a cycle in a graph *Forest (graph theory), an undirected graph with no cycles *Biconnected graph, an undirected graph in which every edge belongs to a cycle *Directed acyclic graph, a directed graph with no cycles *Strongly connected graph, a directed graph in which every edge belongs to a cycle *Aperiodic graph, a directed graph in which the cycle lengths have no nontrivial common divisor *Pseudoforest, a directed or undirected graph in which every connected component includes at most one cycle *Cycle graph, a graph that has the structure of a single cycle *Pancyclic graph, a graph that has cycles of all possible lengths *Cycle detection (graph theory), the algorithmic problem of finding cycles in graphs Other similarly-named concepts include *Cycle graph (algebra), a graph that illustrates the cyclic subgroups of a group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconnected Graph
In graph theory, a biconnected graph is a connected and "nonseparable" graph, meaning that if any one vertex were to be removed, the graph will remain connected. Therefore a biconnected graph has no articulation vertices. The property of being 2-connected is equivalent to biconnectivity, except that the complete graph of two vertices is usually not regarded as 2-connected. This property is especially useful in maintaining a graph with a two-fold redundancy, to prevent disconnection upon the removal of a single edge (or connection). The use of biconnected graphs is very important in the field of networking (see Network flow), because of this property of redundancy. Definition A biconnected undirected graph is a connected graph that is not broken into disconnected pieces by deleting any single vertex (and its incident edges). A biconnected directed graph is one such that for any two vertices ''v'' and ''w'' there are two directed paths from ''v'' to ''w'' which have no vert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Acyclic Graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs. Definitions A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Connected Graph
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(''V'' + ''E'')). Definitions A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first. In a directed graph ''G'' that may not itself be strongly connected, a pair of vertices ''u'' and ''v'' are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Graph
In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer ''k'' > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph ''G'' is called the ''period'' of ''G''. Graphs that cannot be aperiodic In any directed bipartite graph, all cycles have a length that is divisible by two. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every ''k'' divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by ''n'', the length of that cycle. Testing for aperiodicity Suppose that ''G'' is strongly connected and that ''k'' divides the lengths of all cycles in ''G''. Consider the results ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoforest
In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest. The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan. attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems.. Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests ... [...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. 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 number of vertices * 2-regular * 2-ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pancyclic Graph
In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph.. Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length. Definitions An ''n''-vertex graph ''G'' is pancyclic if, for every k in the range 3 \leq k \leq n \; ,G contains a cycle of length k. It is node-pancyclic or vertex-pancyclic if, for every vertex ''v'' and every ''k'' in the same range, it contains a cycle of length ''k'' that contains ''v''.. Similarly, it is edge-pancyclic if, for every edge ''e'' and every ''k'' in the same range, it contains a cycle of length ''k'' that contains ''e''. A bipartite graph cannot be pancyclic, because it does not contain any odd-length cycles, but it is said to be bipancyclic if it contains cycles of all even lengths from 4 to ''n''. Planar graphs A maximal outerplanar graph is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Detection (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph (algebra)
In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a given group element ''a'', where ''an'', the ''n''-th power of an element ''a'' is defined as the product of ''a'' multiplied by itself ''n'' times. The element ''a'' is said to ''generate'' the cycle. In a finite group, some non-zero power of ''a'' must be the group identity, ''e''; the lowest such power is the order of the cycle, the number of distinct elements in it. In a cycle graph, the cycle is represented as a polygon, with the vertices representing the group elements, and the connecting lines indicating that all elements in that polygon are members of the same cycle. Cycles Cycles can overlap, or they can have no element in common but the identity. The cycle graph displays each interesting cycle as a polygon. If ''a'' generates a cycle of order 6 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circulant Graph
In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings. Equivalent definitions Circulant graphs can be described in several equivalent ways:. *The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has a graph automorphism, which is a cyclic permutation of its vertices. *The graph has an adjacency matrix that is a circulant matrix. *The vertices of the graph can be numbered from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. *The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing. *The graph is a Cayley graph of a cyclic group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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