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Graph Toughness
In graph theory, toughness is a measure of the connectivity of a graph. A graph is said to be -tough for a given real number if, for every integer , cannot be split into different connected components by the removal of fewer than vertices. For instance, a graph is -tough if the number of components formed by removing a set of vertices is always at most as large as the number of removed vertices. The toughness of a graph is the maximum for which it is -tough; this is a finite number for all finite graphs except the complete graphs, which by convention have infinite toughness. Graph toughness was first introduced by . Since then there has been extensive work by other mathematicians on toughness; the recent survey by lists 99 theorems and 162 papers on the subject. Examples Removing vertices from a path graph can split the remaining graph into as many as connected components. The maximum ratio of components to removed vertices is achieved by removing one vertex (from the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-NP
In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP if and only if for every ''no''-instance we have a polynomial-length " certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate. That is, co-NP is the set of decision problems where there exists a polynomial and a polynomial-time bounded Turing machine ''M'' such that for every instance ''x'', ''x'' is a ''no''-instance if and only if: for some possible certificate ''c'' of length bounded by , the Turing machine ''M'' accepts the pair . Complementary problems While an NP problem asks whether a given instance is a ''yes''-instance, its ''complement'' asks whether an instance is a ''no''-instance, which means the complement is in co-NP. Any ''yes''-instance for the original NP problem becomes a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Connectivity
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two vertices and are called connected if contains a path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length (that is, they are the endpoints of a single edge), the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. An undirected graph is therefore disconnected if there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Applied Mathematics
''Discrete Applied Mathematics'' is a peer-reviewed scientific journal covering algorithmic and applied areas of discrete mathematics. It is published by Elsevier and the editor-in-chief is Endre Boros (Rutgers University). The journal was split off from another Elsevier journal, ''Discrete Mathematics'', in 1979, with that journal's founder Peter Ladislaw Hammer as its founding editor-in-chief. Abstracting and indexing The journal is abstracted and indexing in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 1.139. References External links *{{official website, http://www.journals.elsevier.com/discrete-applied-mathematics/ Discrete mathematics journals Academic journals established in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphs And Combinatorics
''Graphs and Combinatorics'' (ISSN 0911-0119, abbreviated ''Graphs Combin.'') is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. Its editor-in-chief is Katsuhiro Ota of Keio University. The journal was first published in 1985. Its founding editor in chief was Hoon Heng Teh of Singapore, the president of the Southeast Asian Mathematics Society, and its managing editor was Jin Akiyama. Originally, it was subtitled "An Asian Journal". In most years since 1999, it has been ranked as a second-quartile journal in discrete mathematics and theoretical computer science by SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journal and the prestige of the journals where the citations come from. Etymology SCImago ..... References {{reflist Academic journals established in 1985 Combinatorics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harris Graph
In graph theory, a Harris Graph theory, graph is defined as an Eulerian path, Eulerian, Graph toughness, tough, non-Hamiltonian graph. Harris graphs were introduced in 2013 when, at the University of Michigan, Harris Spungen conjectured that any graph which is both Eulerian path, tough and Eulerian is sufficient, sufficiently Hamiltonian. However, Douglas Shaw disproved this conjecture, discovering a counterexample with Order of a graph, order 9 and Size (graph theory), size 14. Currently, there are 241,375 known Harris graphs. The minimal Harris graph, the Hirotaka graph, has order 7 and size 12. Harris graph theory, graphs can be constructed by adding barnacles or grafting smaller Harris graph theory, graphs, enabling larger graphs while preserving their properties. Notable types include the minimal Hirotaka graph theory, graph, the barnacle-free Lopez graph theory, graph, and the Shaw graph theory, graph, each showcasing unique structural features in graph theory. Harris grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte–Berge Formula
In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte theorem on perfect matchings, and is named after W. T. Tutte (who proved Tutte's theorem) and Claude Berge (who proved its generalization). Statement The theorem states that the size of a maximum matching of a graph G=(V,E) equals \frac \min_ \left(, U, -\operatorname(G-U)+, V, \right), where \operatorname(H) counts how many of the connected components of the graph H have an odd number of vertices. Equivalently, the number of ''unmatched'' vertices in a maximum matching equals\max_ \left(\operatorname(G-U)-, U, \right) . Explanation Intuitively, for any subset U of the vertices, the only way to completely cover an odd component of G-U by a matching is for one of the matched edges covering the component to be incident to U. If, instead, some odd component had no matched edge connecting it to U, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strength Of A Graph
In graph theory, the strength of an undirected graph corresponds to the minimum ratio of ''edges removed''/''components created'' in a decomposition of the graph in question. It is a method to compute partitions of the set of vertices and detect zones of high concentration of edges, and is analogous to graph toughness which is defined similarly for vertex removal. Definitions The strength \sigma(G) of an undirected simple graph ''G'' = (''V'', ''E'') admits the three following definitions: * Let \Pi be the set of all partitions of V, and \partial \pi be the set of edges crossing over the sets of the partition \pi\in\Pi, then \displaystyle\sigma(G)=\min_\frac. * Also if \mathcal T is the set of all spanning trees of ''G'', then :: \sigma(G)=\max\left\. * And by linear programming duality, :: \sigma(G)=\min\left\. Complexity Computing the strength of a graph can be done in polynomial time, and the first such algorithm was discovered by Cunningham (1985). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # 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. Hence, 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
