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Penny Graph
In geometric graph theory, a penny graph is a contact graph of unit circles. It is formed from a collection of unit circles that do not cross each other, by creating a vertex for each circle and an edge for every pair of tangent circles. The circles can be represented physically by Penny, pennies, arranged without overlapping on a flat surface, with a vertex for each penny and an edge for each two pennies that touch. Penny graphs have also been called unit coin graphs, because they are the Circle packing theorem, coin graphs formed from unit circles. If each vertex is represented by a point at the center of its circle, then two vertices will be adjacent if and only if their distance is the minimum distance among all pairs of points. Therefore, penny graphs have also been called minimum-distance graphs, smallest-distance graphs, or closest-pairs graphs. Similarly, in a mutual nearest neighbor graph that links pairs of points in the plane that are each other's Nearest neighbor graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penny Graph
In geometric graph theory, a penny graph is a contact graph of unit circles. It is formed from a collection of unit circles that do not cross each other, by creating a vertex for each circle and an edge for every pair of tangent circles. The circles can be represented physically by Penny, pennies, arranged without overlapping on a flat surface, with a vertex for each penny and an edge for each two pennies that touch. Penny graphs have also been called unit coin graphs, because they are the Circle packing theorem, coin graphs formed from unit circles. If each vertex is represented by a point at the center of its circle, then two vertices will be adjacent if and only if their distance is the minimum distance among all pairs of points. Therefore, penny graphs have also been called minimum-distance graphs, smallest-distance graphs, or closest-pairs graphs. Similarly, in a mutual nearest neighbor graph that links pairs of points in the plane that are each other's Nearest neighbor graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Approximation Scheme
In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter and produces a solution that is within a factor of being optimal (or for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most , with being the length of the shortest tour. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time or even counts as a PTAS. Variants Deterministic A practical problem with PTAS algorithms is that the exponent of the polynomial could increase dramatically as ε shrinks, for example if the runtime is . One way of addressing this is to define the efficient polynomial-time a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker's Technique
In theoretical computer science, Baker's technique is a method for designing polynomial-time approximation schemes (PTASs) for problems on planar graphs. It is named after Brenda Baker, who announced it in a 1983 conference and published it in the ''Journal of the ACM'' in 1994. The idea for Baker's technique is to break the graph into layers, such that the problem can be solved optimally on each layer, then combine the solutions from each layer in a reasonable way that will result in a feasible solution. This technique has given PTASs for the following problems: subgraph isomorphism, maximum independent set, minimum vertex cover, minimum dominating set, minimum edge dominating set, maximum triangle matching, and many others. The bidimensionality theory of Erik Demaine, Fedor Fomin, Hajiaghayi, and Dimitrios Thilikos and its offshoot ''simplifying decompositions'' (,) generalizes and greatly expands the applicability of Baker's technique for a vast set of problems on planar grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Disjoint Set
In computational geometry, a maximum disjoint set (MDS) is a largest set of non-overlapping geometric shapes selected from a given set of candidate shapes. Every set of non-overlapping shapes is an independent set in the intersection graph of the shapes. Therefore, the MDS problem is a special case of the maximum independent set (MIS) problem. Both problems are NP complete, but finding a MDS may be easier than finding a MIS in two respects: * For the general MIS problem, the best known exact algorithms are exponential. In some geometric intersection graphs, there are sub-exponential algorithms for finding a MDS. * The general MIS problem is hard to approximate and doesn't even have a constant-factor approximation. In some geometric intersection graphs, there are polynomial-time approximation schemes (PTAS) for finding a MDS. Finding an MDS is important in applications such as automatic label placement, VLSI circuit design, and cellular frequency division multiplexing. The MDS pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grötzsch's Theorem
In the mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors. According to the four-color theorem, every graph that can be drawn in the plane without edge crossings can have its vertices colored using at most four different colors, so that the two endpoints of every edge have different colors, but according to Grötzsch's theorem only three colors are needed for planar graphs that do not contain three mutually adjacent vertices. History The theorem is named after German mathematician Herbert Grötzsch, who published its proof in 1959. Grötzsch's original proof was complex. attempted to simplify it but his proof was erroneous.. In 2003, Carsten Thomassen derived an alternative proof from another related theorem: every planar graph with girth at least five is 3-list-colorable. However, Grötzsch's theorem itself does not extend from coloring to list coloring: there exist triangle-free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle-free Penny Graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with edges is triangle-free in time . Another approach is to find the trace of , where is the adjacency matrix of the graph. The trace is zero if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-color Theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet. It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubters remain. The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples (unlike the five color theorem, proved in the 1800s, which states that five colors are enough to color a map). To dispel any remaining doubts about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was publis ... [...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] |
Degeneracy (graph Theory)
In graph theory, a ''k''-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most ''k'': that is, some vertex in the subgraph touches ''k'' or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of ''k'' for which it is ''k''-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph. Degeneracy is also known as the ''k''-core number, width, and linkage, and is essentially the same as the coloring number or Szekeres–Wilf number (named after ). ''k''-degenerate graphs have also been called ''k''-inductive graphs. The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been (repeatedly) removed are called the ''k''-cores of the graph and the degeneracy of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penny Graph 11 Nodes
A penny is a coin ( pennies) or a unit of currency (pl. pence) in various countries. Borrowed from the Carolingian denarius (hence its former abbreviation d.), it is usually the smallest denomination within a currency system. Presently, it is the formal name of the British penny ( p) and the ''de facto'' name of the American one-cent coin (abbr. ¢) as well as the informal Irish designation of the 1 cent euro coin (abbr. c). It is the informal name of the cent unit of account in Canada, although one-cent coins are no longer minted there. The name is used in reference to various historical currencies, also derived from the Carolingian system, such as the French denier and the German pfennig. It may also be informally used to refer to any similar smallest-denomination coin, such as the euro cent or Chinese fen. The Carolingian penny was originally a 0.940-fine silver coin, weighing pound. It was adopted by Offa of Mercia and other English kings and remained th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle-free Graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with edges is triangle-free in time . Another approach is to find the trace of , where is the adjacency matrix of the graph. The trace is zero if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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