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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 small 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 graphs can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Clique-width Construction
In graph theory, the clique-width of a Graph (discrete mathematics), graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be small for dense graphs. It is defined as the minimum number of Graph labeling, labels needed to construct by means of the following 4 operations : #Creation of a new vertex with label (denoted by ) #Disjoint union of graphs, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graph Power
In graph theory, a branch of mathematics, the th power of an undirected graph is another graph that has the same set of vertex (graph theory), vertices, but in which two vertices are adjacent when their Distance (graph theory), distance in is at most . Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: is called the ''Square number, square'' of , is called the ''Cube (algebra), cube'' of , etc. Graph powers should be distinguished from the Graph product, products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph. Properties If a graph has graph diameter, diameter , then its -th power is the complete graph. If a graph family has bounded clique-width, then so do its -th powers for any fixed . Coloring Graph coloring on the square of a graph may be used to assign frequencies to the participants of wireless communication networks so that no two participants interfere with ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Parameterized Complexity
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to ''multiple'' parameters of the input or output. The complexity of a problem is then measured as a function of those parameters. This allows the classification of NP-hard problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input. This appears to have been first demonstrated in . The first systematic work on parameterized complexity was done by . Under the assumption that P ≠ NP, there exist many natural problems that require super-polynomial running time when complexity is measured in terms of the input size only but that are computable in a time that is polynomial in the input size and exponential or worse in a parameter . Hence, if is fixed at a small value and the growth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Split Decomposition
In graph theory, a split of an undirected graph is a Cut (graph theory), cut whose cut-set forms a complete bipartite graph. A graph is prime if it has no splits. The splits of a graph can be collected into a tree-like structure called the split decomposition or join decomposition, which can be constructed in linear time. This decomposition has been used for fast recognition of circle graphs and distance-hereditary graphs, as well as for other problems in graph algorithms. Splits and split decompositions were first introduced by , who also studied variants of the same notions for directed graphs.. Definitions A Cut (graph theory), cut of an undirected graph is a partition of the vertices into two nonempty subsets, the sides of the cut. The subset of edges that have one endpoint in each side is called a cut-set. When a cut-set forms a complete bipartite graph, its cut is called a split. Thus, a split can be described as a partition of the vertices of the graph into two subsets an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 chro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hamiltonian Cycle
In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle (graph theory), cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details. Hamiltonian paths and cycles are named after William Rowan Hamilton, who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graph Coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the Vertex (graph theory), 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 (graph theory), edges 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 (graph theory), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Monadic Second-order Logic
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order formulas over graphs of bounded treewidth. It is also of fundamental importance in automata theory, where the Büchi–Elgot–Trakhtenbrot theorem gives a logical characterization of the regular languages. Second-order logic allows quantification over Predicate (mathematical logic), predicates. However, MSO is the Fragment (logic), fragment in which second-order quantification is limited to monadic predicates (predicates having a single argument). This is often described as quantification over "sets" because monadic predicates are equivalent in expressive power to sets (the set of elements for which the predicate is true). Variants Monadic second-order logic come ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graph Property
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph.. Definitions While graph drawing and graph representation are valid topics in graph theory, in order to focus only on the abstract structure of graphs, a graph property is defined to be a property preserved under all possible isomorphisms of a graph. In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph. Informally, the term "graph invariant" is used for properties expressed quantitatively, while "property" usually refers to descriptive characterizations of graphs. For example, the statement "graph does not have vertices of degree 1" is a "property" while "the number of vertices of degree 1 in a graph" is an "invariant". More formally, a graph property is a class of graphs with the property that any two isomorphic gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rank-width
Rank-width is a graph width parameter used in graph theory and parameterized complexity, and defined using linear algebra. It is defined from hierarchical clusterings of the vertices of a given graph, which can be visualized as ternary trees having the vertices as their leaves. Removing any edge from such a tree disconnects it into two subtrees and partitions the vertices into two subsets. The graph edges that cross from one side of the partition to the other can be described by a biadjacency matrix; for the purposes of rank-width, this matrix is defined over the finite field GF(2) rather than using real numbers. The rank-width of a graph is the maximum of the ranks of the biadjacency matrices, for a clustering chosen to minimize this maximum. Rank-width is closely related to clique-width: k \leq c \leq 2^-1, where c is the clique-width and k the rank-width. However, clique-width is NP-hard to compute, for graphs of large clique-width, and its parameterized complexity is unknown. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sparse Graph
In mathematics, a dense graph is a Graph (discrete mathematics), graph in which the number of edges is close to the maximal number of edges (where every pair of Vertex (graph theory), vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem. The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges. For undirected simple graphs, the graph density is: :D = \frac = \frac For Directed graph, directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: :D = \frac = \frac where is the number of edges and is the number of vertices in the graph. The maximum number of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |