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Glossary Of Graph Theory
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by .... Symbols A B C D E F G H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by ''edges'' (also called ''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 of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In contrast, the unordered pair 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 embedding of a graph to embeddings into a ''book'', a collection of half-planes all having the same 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 subhamiltonian graphs, which are always planar; more generally, ... [...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 graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Graph
In graph theory, a branch of combinatorial mathematics, a block graph or clique tree. is a type of undirected graph in which every biconnected component (block) is a clique. Block graphs are sometimes erroneously called Husimi trees (after Kôdi Husimi), but that name more properly refers to cactus graphs, graphs in which every nontrivial biconnected component is a cycle. Block graphs may be characterized as the intersection graphs of the blocks of arbitrary undirected graphs.. Characterization Block graphs are exactly the graphs for which, for every four vertices , , , and , the largest two of the three distances , , and are always equal... They also have a forbidden graph characterization as the graphs that do not have the diamond graph or a cycle of four or more vertices as an induced subgraph; that is, they are the diamond-free chordal graphs. They are also the Ptolemaic graphs (chordal distance-hereditary graphs) in which every two nodes at distance two from each ... [...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 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 contributes the same amount (one) to both numbers. Symmetry Every regular bipartite graph is also biregular. Every edge-transitive graph (disallowing graphs with isolated vertices) that is not also vertex-transitive must be biregular.. In particular every ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a 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 cycles. The two sets U and V may be thought of as a 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 triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denotin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconnected Component
In graph theory, a biconnected component (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. 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 search while maintaining the following information: # the depth of each vertex in the depth-first-search tree (once it gets visited), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
