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Χ-bounded
In graph theory, a \chi-bounded family \mathcal of graphs is one for which there is some function f such that, for every integer t the graphs in \mathcal with t=\omega(G) (clique number) can be colored with at most f(t) colors. The function f(t) is called a \chi-binding function for \mathcal. These concepts and their notations were formulated by András Gyárfás. The use of the Greek letter chi in the term \chi-bounded is based on the fact that the chromatic number of a graph G is commonly denoted \chi(G). An overview of the area can be found in a survey of Alex Scott and Paul Seymour. Nontriviality It is not true that the family of all graphs is \chi-bounded. As , and showed, there exist triangle-free graphs of arbitrarily large chromatic number, so for these graphs it is not possible to define a finite value of f(2). Thus, \chi-boundedness is a nontrivial concept, true for some graph families and false for others. Specific classes Every class of graphs of bounded chromatic nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is p ... [...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] |
Intersection Graph
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them. Formal definition Formally, an intersection graph is an undirected graph formed from a family of sets : S_i, \,\,\, i = 0, 1, 2, \dots by creating one vertex for each set , and connecting two vertices and by an edge whenever the corresponding two sets have a nonempty intersection, that is, : E(G) = \. All graphs are intersection graphs Any undirected graph may be represented as an intersection graph. For each vertex of , form a set consisting of the edges incident to ; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Therefore, is the intersection graph of the sets . provide a construction that is more e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subset V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. * Induced paths are induced subgraphs that are paths. The shortest path betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is call ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyárfás–Sumner Conjecture
In graph theory, the Gyárfás–Sumner conjecture asks whether, for every tree T and complete graph K, the graphs with neither T nor K as induced subgraphs can be properly colored using only a constant number of colors. Equivalently, it asks whether the T-free graphs are \chi-bounded. It is named after András Gyárfás and David Sumner, who formulated it independently in 1975 and 1981 respectively. It remains unproven. In this conjecture, it is not possible to replace T by a graph with cycles. As Paul Erdős and András Hajnal have shown, there exist graphs with arbitrarily large chromatic number and, at the same time, arbitrarily large girth. Using these graphs, one can obtain graphs that avoid any fixed choice of a cyclic graph and clique (of more than two vertices) as induced subgraphs, and exceed any fixed bound on the chromatic number. The conjecture is known to be true for certain special choices of T, including paths, stars, and trees of radius two. It is also known tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as \overline). Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a chord (of that curve). In real or complex vector spaces If ''V'' is a vector spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Graph
In 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 conne ..., a string graph is an intersection graph of Plane curve, curves in the plane; each curve is called a "string". Given a Graph (discrete mathematics), graph , is a string graph if and only if there exists a set of curves, or strings, such that the graph having a Vertex (graph theory), vertex for each curve and an edge for each intersecting pair of curves is Graph isomorphism, isomorphic to . Background described a concept similar to string graphs as they applied to genetic structures. In that context, he also posed the specific case of intersecting intervals on a line, namely the now classical family of interval graphs. Later, specified the same idea to electrical networks and printed circuits. The mathematical st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even-hole-free Graph
In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices. More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs. demonstrated that every even-hole-free graph contains a bisimplicial vertex, which settled a conjecture by Reed. Recognition gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in (n^) time. later improved this to (n^). and improved this to (n^) time. The best currently known algorithm is given by which runs in (n^9) time. While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex. It is unknown whether graph coloring and the maximum independent set problem can be solved in polynomial time on even-hole-free graphs, or whether the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrangement Of Lines
In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestration in that the latter process is limited to the assignment of notes to instruments for performance by an orchestra, concert band, or other musical ensemble. Arranging "involves adding compositional techniques, such as new thematic material for introductions, transitions, or modulations, and endings. Arranging is the art of giving an existing melody musical variety".(Corozine 2002, p. 3) In jazz, a memorized (unwritten) arrangement of a new or pre-existing composition is known as a ''head arrangement''. Classical music Arrangement and transcriptions of classical and serious music go back to the early history of this genre. Eighteenth century J.S. Bach frequently made arrangements of his own and other composers' pieces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
