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Matching Preclusion
In graph theory, a branch of mathematics, the matching preclusion number of a graph ''G'' (denoted mp(''G'')) is the minimum number of edges whose deletion results in the elimination of all perfect matchings or near-perfect matchings (matchings that cover all but one vertex in a graph with an odd number of vertices). Matching preclusion measures the robustness of a graph as a communications network topology for distributed algorithms that require each node of the distributed system to be matched with a neighboring partner node. In many graphs, mp(''G'') is equal to the minimum degree of any vertex in the graph, because deleting all edges incident to a single vertex prevents that vertex from being matched. This set of edges is called a trivial matching preclusion set.. A variant definition, the conditional matching preclusion number, asks for the minimum number of edges the deletion of which results in a graph that has neither a perfect or near-perfect matching nor any isolated ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph with edges and vertices , a perfect matching in is a subset of , such that every vertex in is adjacent to exactly one edge in . The adjacency matrix of a perfect matching is a symmetric permutation matrix. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs:Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5. : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications Network
A telecommunications network is a group of nodes interconnected by telecommunications links that are used to exchange messages between the nodes. The links may use a variety of technologies based on the methodologies of circuit switching, message switching, or packet switching, to pass messages and signals. Multiple nodes may cooperate to pass the message from an originating node to the destination node, via multiple network hops. For this routing function, each node in the network is assigned a network address for identification and locating it on the network. The collection of addresses in the network is called the address space of the network. Examples of telecommunications networks include computer networks, the Internet, the public switched telephone network (PSTN), the global Telex network, the aeronautical ACARS network, and the wireless radio networks of cell phone telecommunication providers. Network structure this is the structure of network general, every telecommu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributed Algorithm
A distributed algorithm is an algorithm designed to run on computer hardware constructed from interconnected processors. Distributed algorithms are used in different application areas of distributed computing, such as telecommunications, scientific computing, distributed information processing, and real-time process control. Standard problems solved by distributed algorithms include leader election, consensus, distributed search, spanning tree generation, mutual exclusion, and resource allocation. Distributed algorithms are a sub-type of parallel algorithm, typically executed concurrently, with separate parts of the algorithm being run simultaneously on independent processors, and having limited information about what the other parts of the algorithm are doing. One of the major challenges in developing and implementing distributed algorithms is successfully coordinating the behavior of the independent parts of the algorithm in the face of processor failures and unreliable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...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] |
Edge Connectivity
In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If the subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is said to be ''k''-edge-connected. The edge connectivity of is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a in ''G''. The edge connectivity version of |
Cyclomatic Number
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 graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Invariants
Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discrete mathematics *Graph of a function * Graph of a relation * Graph paper *Chart, a means of representing data (also called a graph) Computing *Graph (abstract data type), an abstract data type representing relations or connections * graph (Unix), Unix command-line utility * Conceptual graph, a model for knowledge representation and reasoning * Microsoft Graph, a Microsoft API developer platform that connects multiple services and devices Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also * Complex network *Graf * Graff (other) * Graph database *Grapheme, in linguistics * Graphemics * Graphic (other) *-graphy The English suffix -graphy means a "field of study" or related to "writing" a book, and is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
