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Knot (graph Theory)
This is a glossary of graph theory. Graph theory is the study of graph (discrete mathematics), graphs, systems of nodes or vertex (graph theory), vertices connected in pairs by lines or #edge, edges. Symbols A B C D E F G H I J K L M ... [...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] |
Ordered Pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unordered pair'', denoted , always equals the unordered pair . Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Book Embedding
In graph theory, a book embedding is a generalization of planar graph, planar embedding of a Graph (discrete mathematics), graph to embeddings in a ''book'', a collection of half-planes all having the same Line (geometry), line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the ''spine'', and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number. Every graph with vertices has book thickness at most \lceil n/2\rceil, and this formula gives the exact book thickness for complete graphs. The graphs with book thickness one are the outerplanar graphs. The graphs with book thickness at most two are the subhamilt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Book (graph Theory)
In graph theory, a book graph (often written B_p ) may be any of several kinds of graph formed by multiple cycles sharing an edge. Variations One kind, which may be called a quadrilateral book, consists of ''p'' quadrilaterals sharing a common edge (known as the "spine" or "base" of the book). That is, it is a Cartesian product of a star and a single edge. The 7-page book graph of this type provides an example of a graph with no harmonious labeling. A second type, which might be called a triangular book, is the complete tripartite graph ''K''1,1,''p''. It is a graph consisting of p triangles sharing a common edge. A book of this type is a split graph. This graph has also been called a K_e(2,p) or a thagomizer graph (after thagomizers, the spiked tails of stegosaurian dinosaurs, because of their pointy appearance in certain drawings) and their graphic matroids have been called thagomizer matroids. Triangular books form one of the key building blocks of line perfect gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Graph
Block or blocked may refer to: Arts, entertainment and media Broadcasting * Block programming, the result of a programming strategy in broadcasting * W242BX, a radio station licensed to Greenville, South Carolina, United States known as ''96.3 the Block '' * WFNZ-FM, a radio station licensed to Harrisburg, North Carolina, United States, branded as ''92.7 The Block'' * "Blocked", an episode of the television series '' The Flash'' Music * Block Entertainment, a record label * Blocks Recording Club, a record label * Woodblock (instrument), a small piece of slit drum made from one piece of wood and used as a percussion instrument * "Blocks", by C418 from '' Minecraft – Volume Beta'', 2013 Toys * Toy block, one of a set of wooden or plastic pieces, of various shapes * Unit block, a type of standardized wooden toy block for children Video games * Blocked (video game), a puzzle game for the iPhone and iPod Touch Building and construction * Concrete block, cinder block or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biregular Graph
In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph G=(U,V,E) for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in U is x and the degree of the vertices in V is y, then the graph is said to be (x,y)-biregular. Example Every complete bipartite graph K_ is (b,a)-biregular. The rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ... is another example; it is (3,4)-biregular. Vertex counts An (x,y)-biregular graph G=(U,V,E) must satisfy the equation x, U, =y, V, . This follows from a simple double counting argument: the number of endpoints of edges in U is x, U, , the number of endpoints of edges in V is y, V, , and each edge contr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconnected Component
In graph theory, a biconnected component or block (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph. The blocks are attached to each other at shared vertices called cut vertices or separating vertices or articulation points. Specifically, a cut vertex is any vertex whose removal increases the number of connected components. A block containing at most one cut vertex is called a leaf block, it corresponds to a leaf vertex in the block-cut tree. Algorithms Linear time depth-first search The classic sequential algorithm for computing biconnected components in a connected undirected graph is due to John Hopcroft and Robert Tarjan (1973). It runs in linear time, and is based on depth-first search. This algorithm is also outlined as Problem 22-2 of Introduction to Algorithms (both 2nd and 3rd editions). The idea is to run a depth-first sea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected Graph (discrete mathematics), graph is said to be -vertex-connected (or -connected) if it has more than Vertex (graph theory), vertices and remains Connectivity (graph theory), connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. In complete graphs, there is no subset whose removal would disconnect the graph. Some sources modify the definition of connectivity to handle this case, by defining it as the size of the smallest subset of vertices whose deletion results in either a disconnected graph or a single vertex. For this variation, the connectivity of a complete graph K_n is n-1. An equivalent definition is that a graph with at least two vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Bandwidth
In graph theory, the graph bandwidth problem is to label the vertices of a graph with distinct integers so that the quantity \max\ is minimized ( is the edge set of ). The problem may be visualized as placing the vertices of a graph at distinct integer points along the ''x''-axis so that the length of the longest edge is minimized. Such placement is called linear graph arrangement, linear graph layout or linear graph placement. The weighted graph bandwidth problem is a generalization wherein the edges are assigned weights and the cost function to be minimized is \max\. In terms of matrices, the (unweighted) graph bandwidth is the minimal bandwidth of a symmetric matrix which is an adjacency matrix of the graph. The bandwidth may also be defined as one less than the maximum clique size in a proper interval supergraph of the given graph, chosen to minimize its clique size . Cyclically interval graphs For fixed k define for every i the set I_k(i) := [i, i+k+1). G_k(n) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge– vertex connectivity. Formally, an automorphism of a graph is a permutation of the vertex set , such that the pair of vertices form an edge if and only if the pair also form an edge. That is, it is a graph isomorphism from to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. Computational complexity Constructing the automorphism group of a graph, in the form of a list of generators, is polynomial-time equivalent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
