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Word-representable Graph
In the mathematical field of graph theory, a word-representable graph is a graph that can be characterized by a word (or sequence) whose entries alternate in a prescribed way. In particular, if the vertex set of the graph is ''V'', one should be able to choose a word ''w'' over the alphabet ''V'' such that letters ''a'' and ''b'' alternate in ''w'' if and only if the pair ''ab'' is an edge in the graph. (Letters ''a'' and ''b'' alternate in ''w'' if, after removing from ''w'' all letters but the copies of ''a'' and ''b'', one obtains a word ''abab''... or a word ''baba''....) For example, the cycle graph labeled by ''a'', ''b'', ''c'' and ''d'' in clock-wise direction is word-representable because it can be represented by ''abdacdbc'': the pairs ''ab'', ''bc'', ''cd'' and ''ad'' alternate, but the pairs ''ac'' and ''bd'' do not. The word ''w'' is ''G'''s ''word-representant'', and one says that that ''w'' ''represents'' ''G''. The smallest (by the number of vertices) non-wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word-representable Graph
In the mathematical field of graph theory, a word-representable graph is a graph that can be characterized by a word (or sequence) whose entries alternate in a prescribed way. In particular, if the vertex set of the graph is ''V'', one should be able to choose a word ''w'' over the alphabet ''V'' such that letters ''a'' and ''b'' alternate in ''w'' if and only if the pair ''ab'' is an edge in the graph. (Letters ''a'' and ''b'' alternate in ''w'' if, after removing from ''w'' all letters but the copies of ''a'' and ''b'', one obtains a word ''abab''... or a word ''baba''....) For example, the cycle graph labeled by ''a'', ''b'', ''c'' and ''d'' in clock-wise direction is word-representable because it can be represented by ''abdacdbc'': the pairs ''ab'', ''bc'', ''cd'' and ''ad'' alternate, but the pairs ''ac'' and ''bd'' do not. The word ''w'' is ''G'''s ''word-representant'', and one says that that ''w'' ''represents'' ''G''. The smallest (by the number of vertices) non-wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Graph
In graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other. Algorithmic complexity gives an O(''n''2)-time algorithm that tests whether a given ''n''-vertex undirected graph is a circle graph and, if it is, constructs a set of chords that represents it. A number of other problems that are NP-complete on general graphs have polynomial time algorithms when restricted to circle graphs. For instance, showed that the treewidth of a circle graph can be determined, and an optimal tree decomposition constructed, in O(''n''3) time. Additionally, a minimum fill-in (that is, a chordal graph with as few edges as possible that contains the given circle graph as a subgraph) may be found in O(''n''3) time. has shown that a maximum clique of a circle graph can be found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crown Graph
In graph theory, a branch of mathematics, a crown graph on vertices is an undirected graph with two sets of vertices and and with an edge from to whenever . The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product , as the complement of the Cartesian direct product of and , or as a bipartite Kneser graph representing the 1-item and -item subsets of an -item set, with an edge between two subsets whenever one is contained in the other. Examples The 6-vertex crown graph forms a cycle, and the 8-vertex crown graph is isomorphic to the graph of a cube. In the Schläfli double six, a configuration of 12 lines and 30 points in three-dimensional space, the twelve lines intersect each other in the pattern of a 12-vertex crown graph. Properties The number of edges in a crown graph is the pronic number . Its achromatic number is : one c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1307
Year 1307 ( MCCCVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. Events By place Europe * October 13 – King Philip IV (the Fair) orders the arrest of the Knights Templar in France. The Templars, together with their Grand Master Jacques de Molay, are imprisoned, interrogated, and tortured into confessing heresy. In Paris, the king's inquisitors torture some 140 Templars, most of whom eventually make confessions. Many are subjected to "fire torture": their legs are fastened in an iron frame and the soles of their feet are greased with fat or butter. Unable to withstand these tortures, many Templars eventually confess. * Januli I da Corogna seizes the Aegean Island of Sifnos and becomes an autonomous lord, by renouncing his allegiance to the Knights Hospitaller. England * Spring – King Robert I (the Bruce) crosses with a small force (some 600 men) from the Isle of Arran in the Firth of Clyde to his e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheel Graph
A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be moved easily facilitating movement or transportation while supporting a load, or performing labor in machines. Wheels are also used for other purposes, such as a ship's wheel, steering wheel, potter's wheel, and flywheel. Common examples are found in transport applications. A wheel reduces friction by facilitating motion by rolling together with the use of axles. In order for wheels to rotate, a moment needs to be applied to the wheel about its axis, either by way of gravity or by the application of another external force or torque. Using the wheel, Sumerians invented a device that spins clay as a potter shapes it into the desired object. Terminology The English word ''wheel'' comes from the Old English word , from Proto-Germanic , fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-completeness
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 deter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP (complexity)
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.''Polynomial time'' refers to how quickly the number of operations needed by an algorithm, relative to the size of the problem, grows. It is therefore a measure of efficiency of an algorithm. An equivalent definition of NP is the set of decision problems ''solvable'' in polynomial time by a nondeterministic Turing machine. This definition is the basis for the abbreviation NP; " nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess abou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Problem
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size. The clique problem arises in the following real-world setting. Consider a social network, where the graph's vertices represent people, and the graph's edges represent mutual acquaintance. Then a clique represents a subset of people who all know each other, and algorithms for finding cliques can be used to discover these groups of m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighbourhood (graph Theory)
In graph theory, an adjacent vertex of a vertex in a graph is a vertex that is connected to by an edge. The neighbourhood of a vertex in a graph is the subgraph of induced by all vertices adjacent to , i.e., the graph composed of the vertices adjacent to and all edges connecting vertices adjacent to . The neighbourhood is often denoted or (when the graph is unambiguous) . The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include itself, and is more specifically the open neighbourhood of ; it is also possible to define a neighbourhood in which itself is included, called the closed neighbourhood and denoted by . When stated without any qualification, a neighbourhood is assumed to be open. Neighbourhoods may be used to represent graphs in computer algorithms, via the adjacency list and adjacency matrix representations. Neighbourhoods are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Minor
In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph nor the complete bipartite graph ., p. 77; . The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions., theorem 4, p. 78; . For every fixed graph , it is possible to test whether is a minor of an input graph in polynomial time; together with the forbidden minor characterization this implies that every graph property preserved by deletions and contractions may be recognized in polynomial time. Other results and conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have as a minor may be formed by glui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism (graph Theory)
In graph theory, two graphs G and G' are homeomorphic if there is a graph isomorphism from some subdivision of G to some subdivision of G'. If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the topological sense. Subdivision and smoothing In general, a subdivision of a graph ''G'' (sometimes known as an expansion) is a graph resulting from the subdivision of edges in ''G''. The subdivision of some edge ''e'' with endpoints yields a graph containing one new vertex ''w'', and with an edge set replacing ''e'' by two new edges, and . For example, the edge ''e'', with endpoints : can be subdivided into two edges, ''e''1 and ''e''2, connecting to a new vertex ''w'': The reverse operation, smoothing out or smoothing a vertex ''w'' with regards to the pair of edges (''e''1, ''e''2) inciden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
