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Polytree
In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree; . or singly connected network.) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph. The term ''polytree'' was coined in 1987 by Rebane and Pearl.. Related structures * An arborescence is a directed rooted tree, i.e. a directed acyclic graph in which there exists a single source node that has a unique path to every other node. Every arborescence is a polytree, but not every polytree is an arborescence. * A multitree is a dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytree
In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree; . or singly connected network.) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph. The term ''polytree'' was coined in 1987 by Rebane and Pearl.. Related structures * An arborescence is a directed rooted tree, i.e. a directed acyclic graph in which there exists a single source node that has a unique path to every other node. Every arborescence is a polytree, but not every polytree is an arborescence. * A multitree is a dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sumner's Conjecture
Sumner's conjecture (also called Sumner's universal tournament conjecture) states that every orientation of every n-vertex tree is a subgraph of every (2n-2)-vertex tournament. David Sumner, a graph theorist at the University of South Carolina, conjectured in 1971 that tournaments are universal graphs for polytrees. The conjecture was proven for all large n by Daniela Kühn, Richard Mycroft, and Deryk Osthus. Examples Let polytree P be a star K_, in which all edges are oriented outward from the central vertex to the leaves. Then, P cannot be embedded in the tournament formed from the vertices of a regular 2n-3-gon by directing every edge clockwise around the polygon. For, in this tournament, every vertex has indegree and outdegree equal to n-2, while the central vertex in P has larger outdegree n-1. Thus, if true, Sumner's conjecture would give the best possible size of a universal graph for polytrees. However, in every tournament of 2n-2 vertices, the average outdegree is n-\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), 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 graphs, 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 undirecte ... [...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] |
Glossary Of Graph Theory
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I K L M N O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (graph Theory)
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of and may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. Sumner's conjecture states that every tournament with vertices contains every polytree with vertices. The number of non-isomorphic oriented graphs with vertices (for ) is : 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … . Tournaments are in one-to-one correspondence with complete directed graphs (graphs in which there is a directed edge in one or both directions between every pair of distinct vertices). A complete directed graph can be converted to an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Graph
In mathematics, a universal graph is an infinite graph that contains ''every'' finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado and is now called the Rado graph or random graph. More recent work has focused on universal graphs for a graph family : that is, an infinite graph belonging to ''F'' that contains all finite graphs in . For instance, the Henson graphs are universal in this sense for the -clique-free graphs. A universal graph for a family of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in ; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph so a hypercube can be said to be a universal graph for trees. However it is not the smallest such graph: it is known that there is a universal graph for -vertex trees, with only vertices and edges, and that this is optimal. A construction based on the planar separato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multitree
In combinatorics and Order theory, order-theoretic mathematics, a multitree may describe either of two equivalent structures: a directed acyclic graph (DAG) in which there is at most one directed path between any two Vertex (graph theory), vertices, or equivalently in which the Glossary of graph theory#subgraph, subgraph reachable from any vertex induces an Tree (graph theory), undirected tree, or a partially ordered set (poset) that does not have four items , , , and forming a diamond suborder with and but with and incomparable to each other (also called a diamond-free poset.). In computational complexity theory, multitrees have also been called strongly unambiguous graphs or mangroves; they can be used to model nondeterministic algorithms in which there is at most one computational path connecting any two states. Multitrees may be used to represent multiple overlapping Taxonomy (general), taxonomies over the same ground set. If a family tree may contain multiple marriage ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Sumner
David P. Sumner is an American mathematician known for his research in graph theory. He formulated Sumner's conjecture that tournaments are universal graphs for polytrees in 1971, and showed in 1974 that all claw-free graphs with an even number of vertices have perfect matchings. He and András Gyárfás independently formulated the Gyárfás–Sumner conjecture according to which, for every tree ''T'', the ''T''-free graphs are χ-bounded. Sumner earned his doctorate from the University of Massachusetts Amherst The University of Massachusetts Amherst (UMass Amherst, UMass) is a public research university in Amherst, Massachusetts and the sole public land-grant university in Commonwealth of Massachusetts. Founded in 1863 as an agricultural college, ... in 1970, under the supervision of David J. Foulis. He is a distinguished professor emeritus at the University of South Carolina.. References External linksHome page Year of birth missing (living people) Living peo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contour Tree
A Reeb graphY. Shinagawa, T.L. Kunii, and Y.L. Kergosien, 1991. Surface coding based on Morse theory. IEEE Computer Graphics and Applications, 11(5), pp.66-78 (named after Georges Reeb by René Thom) is a mathematical object reflecting the evolution of the level sets of a real-valued function on a manifold. According to a similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem. Proposed by G. Reeb as a tool in Morse theory, Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields \psi, \lambda, and \phi arising from the conditions \nabla \psi = \lambda \nabla \phi and \lambda \neq 0, because these relationships are single-valued when restricted to a region associated with an individual edge of the Reeb graph. This general principle was first used to study neutral surfaces in oceanography. Reeb graphs have also found a wide variety of applications in computati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Edge
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Network
A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (''e.g.'' speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams. Graphical mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
