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Hypertree
In the mathematical field of graph theory, a hypergraph is called a hypertree if it admits a host graph such that is a tree. In other words, is a hypertree if there exists a tree such that every hyperedge of is the set of vertices of a connected subtree of . Hypertrees have also been called arboreal hypergraphs or tree hypergraphs. Every tree is itself a hypertree: itself can be used as the host graph, and every edge of is a subtree of this host graph. Therefore, hypertrees may be seen as a generalization of the notion of a tree for hypergraphs. They include the connected Berge-acyclic hypergraphs, which have also been used as a (different) generalization of trees for hypergraphs. Properties Every hypertree has the Helly property (2-Helly property): if a subset of its hyperedges has the property that every two hyperedges in have a nonempty intersection, then itself has a nonempty intersection (a vertex that belongs to all hyperedges in ). By results of Duchet, Fla ...
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Tree (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 call ...
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Chordal Graph
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs. or triangulated graphs.. Chordal graphs are a subset of the perfect graphs. They may be recognized in linear time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it. Perfect elimination and efficient recogn ...
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Journal Of The ACM
The ''Journal of the ACM'' is a peer-reviewed scientific journal covering computer science in general, especially theoretical aspects. It is an official journal of the Association for Computing Machinery. Its current editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The highest-ranking editor of a publication may also be titled editor, managing ... is Venkatesan Guruswami. The journal was established in 1954 and "computer scientists universally hold the ''Journal of the ACM'' in high esteem". See also * '' Communications of the ACM'' References External links * Publications established in 1954 Computer science journals Association for Computing Machinery academic journals Bimonthly journals English-language journals {{compu-journal-stub ...
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Lecture Notes In Computer Science
''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post- proceedings, monographs, and Festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the ...s. In addition, tutorials, state-of-the-art surveys, and "hot topics" are increasingly being included. The series is indexed by DBLP. See also *'' Monographiae Biologicae'', another monograph series published by Springer Science+Business Media *'' Lecture Notes in Physics'' *'' Lecture Notes in Mathematics'' *'' Electronic Workshops in Computing'', published by the British Computer Society References External links * Publications established in 1973 Computer science books Series of non-fiction books ...
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SIAM Journal On Discrete Mathematics
'' SIAM Journal on Discrete Mathematics'' is a peer-reviewed mathematics journal published quarterly by the Society for Industrial and Applied Mathematics (SIAM). The journal includes articles on pure and applied discrete mathematics. It was established in 1988, along with the '' SIAM Journal on Matrix Analysis and Applications'', to replace the '' SIAM Journal on Algebraic and Discrete Methods''. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.57. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor of 0.755. Although its official ISO abbreviation is ''SIAM J. Discrete Math.'', its publisher and contributors frequently use the shorter abbreviation ''SIDMA''. References External links * Combinatorics journals Publications established in 1988 English-language journals Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a wa ...
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Dually Chordal Graph
In the mathematical area of graph theory, an undirected graph is dually chordal if the hypergraph of its maximal cliques is a hypertree. The name comes from the fact that a graph is chordal if and only if the hypergraph of its maximal cliques is the dual of a hypertree. Originally, these graphs were defined by maximum neighborhood orderings and have a variety of different characterizations. Unlike for chordal graphs, the property of being dually chordal is not hereditary, i.e., induced subgraphs of a dually chordal graph are not necessarily dually chordal (hereditarily dually chordal graphs are exactly the strongly chordal graphs), and a dually chordal graph is in general not a perfect graph. Dually chordal graphs appeared first under the name HT-graphs. Characterizations Dually chordal graphs are the clique graphs of chordal graphs, i.e., the intersection graphs of maximal cliques of chordal graphs. The following properties are equivalent: *''G'' has a maximum neighborhoo ...
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NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a det ...
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Exact Cover
In the mathematical field of combinatorics, given a collection of subsets of a set , an exact cover is a subcollection of such that each element in is contained in ''exactly one'' subset in . In other words, is a partition of consisting of subsets contained in . One says that each element in is covered by exactly one subset in . An exact cover is a kind of cover. In computer science, the exact cover problem is a decision problem to determine if an exact cover exists. The exact cover problem is NP-complete This book is a classic, developing the theory, then cataloguing ''many'' NP-Complete problems. and is one of Karp's 21 NP-complete problems. It is NP-complete even when each subset in contains exactly three elements; this restricted problem is known as exact cover by 3-sets, often abbreviated X3C. The exact cover problem is a kind of constraint satisfaction problem. An exact cover problem can be represented by an incidence matrix or a bipartite graph. Knuth's Al ...
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Linear 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 ...
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Alpha-acyclic
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) where X is a set of elements called ''nodes'' or ''vertices'', and E is a set of non-empty subsets of X called ''hyperedges'' or ''edges''. Therefore, E is a subset of \mathcal(X) \setminus\, where \mathcal(X) is the power set of X. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''. A directed hypergraph differs in that its hyperedges are not sets, but ordered pairs of subsets of X, with each pair's first and second entries constituting the tail and head of the hyperedge respectively. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same car ...
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Conformal Hypergraph
Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph. Clique complex The clique complex of an undirected graph is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of . Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family. The clique complex can also be viewed as a topological space in which each clique of vertices is represented by a simplex of dimension . The 1-skeleton of (also known as the ''underlying graph'' of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2 ...
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