|
χ-bounded
In graph theory, a \chi-bounded family \mathcal of graphs is one for which there is some function c such that, for every integer t the graphs in \mathcal with t=\omega(G) (clique number) can be colored with at most c(t) colors. This concept and its notation 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). 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 t(3). 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 number is (trivially) \chi-bounded, with c(t) equal to the bound on the chromatic number. This includes, for instance, the planar graphs, the bipartite gr ... [...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] |
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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique-width
In graph theory, the clique-width of a graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be bounded even for dense graphs. It is defined as the minimum number of labels needed to construct by means of the following 4 operations : #Creation of a new vertex with label (denoted by ) #Disjoint union of two labeled graphs and (denoted by G \oplus H) #Joining by an edge every vertex labeled to every vertex labeled (denoted by ), where #Renaming label to label (denoted by ) Graphs of bounded clique-width include the cographs and distance-hereditary graphs. Although it is NP-hard to compute the clique-width when it is unbounded, and unknown whether it can be computed in polynomial time when it is bounded, efficient approximation algorithms for clique-width are known. Based on these algorithms and on Courcelle's theorem, many graph optimization problems that are NP-hard for arbitrary graph ... [...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] |
András Gyárfás
András Gyárfás (born 1945) is a Hungarian mathematician who specializes in the study of graph theory. He is famous for two conjectures: * Together with Paul Erdős he conjectured what is now called the Erdős–Gyárfás conjecture which states that any graph with minimum degree 3 contains a simple cycle whose length is a power of two. * He and David Sumner independently formulated the Gyárfás–Sumner conjecture according to which, for every tree ''T'', the ''T''-free graphs are χ-bounded. Gyárfás began working as a researcher for the Computer and Automation Research Institute of the Hungarian Academy of Sciences in 1968. He earned a candidate degree in 1980, and a doctorate (Dr. Math. Sci.) in 1992. He won the Géza Grünwald Commemorative Prize for young researchers of the János Bolyai Mathematical Society in 1978. He was co-author with Paul Erdős on 15 papers, and thus has Erdős number The Erdős number () describes the "collaborative distance" betwee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claw-free Graph
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph ''K''1,3 (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, and the characterization of claw-free perfect graphs., p. 88. They are the subject of hundreds ... [...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] |
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] |
Tree (graph Theory)
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 tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), 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 graphs, 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 undirecte ... [...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] |
