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Simultaneous Embedding
Simultaneous embedding is a technique in graph drawing and information visualization for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between an edge of one graph and an edge of the other graph are allowed. If edges are allowed to be drawn as polylines or curves, then any planar graph may be drawn without crossing with its vertices in arbitrary positions in the plane, where the same vertex placement provides a simultaneous embedding. There are two restricted models: simultaneous geometric embedding, where each graph must be drawn planarly with line segments representing its edges rather than more complex curves, restricting the two given graphs to subclasses of the planar graphs, and simultaneous embedding with fixed edges, where curves or bends are allowed in the edges, but any edge in both graphs must be represented by the same curve in both drawings. In the unrestricted mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graph (discrete mathematics), graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertex (graph theory), vertices and edge (graph theory), edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's menta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star (graph Theory)
In graph theory, a star is the complete bipartite graph a tree with one internal node and leaves (but no internal nodes and leaves when ). Alternatively, some authors define to be the tree of order with maximum diameter 2; in which case a star of has leaves. A star with 3 edges is called a claw. The star is edge-graceful when is even and not when is odd. It is an edge-transitive matchstick graph, and has diameter 2 (when ), girth ∞ (it has no cycles), chromatic index , and chromatic number 2 (when ). Additionally, the star has large automorphism group, namely, the symmetric group on letters. Stars may also be described as the only connected graphs in which at most one vertex has degree greater than one. Relation to other graph families Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-completeness
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a deter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent. Two notable examples in mathematics that have been solved and ''closed'' by researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem.K. Appel and W. Haken (1977), "Every planar map is four colorable. Part I. Discharging", ''Illinois J. Math'' 21: 429–490. K. Appel, W. Haken, and J. Koch (1977), "Every planar map is four colorable. Part II. Reducibility", ''Illinois J. Math'' 21: 491–567. An important open mathematics problem solved i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Forest
In graph theory, a branch of mathematics, a linear forest is a kind of forest formed from the disjoint union of path graphs. It is an undirected graph with no cycles in which every vertex has degree at most two. Linear forests are the same thing as claw-free forests. They are the graphs whose Colin de Verdière graph invariant is at most 1. The linear arboricity of a graph is the minimum number of linear forests into which the graph can be partitioned. For a graph of maximum degree \Delta, the linear arboricity is always at least \lceil\Delta/2\rceil, and it is conjectured that it is always at most \lfloor(\Delta+1)/2\rfloor. A linear coloring of a graph is a proper graph coloring in which the 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. Defini ... formed by each two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outerplanar Graph
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors and , or by their Colin de Verdière graph invariants. They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2. The outerplanar graphs are a subset of the planar graphs, the subgraphs of series–parallel graphs, and the circle graphs. The maximal outerplanar graphs, those to which no more edges can be added while preserving outerplanarity, are also chordal graphs and visibility graphs. History Outerplanar graphs were first studied and named by , in connection with the problem of determining the planarity of graphs formed by using a perfect matching to connect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symposium On Computational Geometry
The International Symposium on Computational Geometry (SoCG) is an academic conference in computational geometry. It was founded in 1985, and was originally sponsored by the SIGACT and SIGGRAPH Special Interest Groups of the Association for Computing Machinery (ACM). It dissociated from the ACM in 2014, motivated by the difficulties of organizing ACM conferences outside the United States and by the possibility of turning to an open-access system of publication. Since 2015 the conference proceedings have been published by the Leibniz International Proceedings in Informatics Dagstuhl is a computer science research center in Germany, located in and named after a district of the town of Wadern, Merzig-Wadern, Saarland. Location Following the model of the mathematical center at Oberwolfach, the center is installed in ... instead of by the ACM. Since 2019 the conference has been organized under the auspices of the newly-formed Society for Computational Geometry. A 2010 assessment of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Theory Of The Reals
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form \exists X_1 \cdots \exists X_n \, F(X_1,\dots, X_n), where the variables X_i are interpreted as having real number values, and where F(X_1,\dots X_n) is a quantifier-free formula involving equalities and inequalities of real polynomials. A sentence of this form is true if it is possible to find values for all of the variables that, when substituted into formula F, make it become true.. The decision problem for the existential theory of the reals is the problem of finding an algorithm that decides, for each such sentence, whether it is true or false. Equivalently, it is the problem of testing whether a given semialgebraic set is non-empty. This decision problem is NP-hard and lies in PSPACE. Thus it has significantly lower complexity than Alfred Tarski's quantifier elimination procedure for deciding statements in the first-or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Symposium On Graph Drawing
The International Symposium on Graph Drawing (GD) is an annual academic conference in which researchers present peer reviewed papers on graph drawing, information visualization of network information, geometric graph theory, and related topics. Significance The Graph Drawing symposia have been central to the growth and development of graph drawing as a research area: as Herman et al. write, "the Graph Drawing community grew around the yearly Symposia." Nguyen lists Graph Drawing as one of "several good conferences which directly or indirectly concern with information visualization", and Wong et al. report that its proceedings "provide a wealth of information". In a 2003 study the symposium was among the top 30% of computer science research publication venues, ranked by impact factor. History The first symposium was held in Marino, near Rome, Italy, in 1992, organized by Giuseppe Di Battista, Peter Eades Peter D. Eades (born 8 January 1952) is an Australian computer scientist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem. A more precise specification is: a problem ''H'' is NP-hard when every problem ''L'' in NP can be reduced in polynomial time to ''H''; that is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve any NP-hard problem would give polynomial time algorithms for all the problems in NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists. It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven. Moreover, the class P, in which all problems can be solved in polynomial time, is contained in the NP class. Defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Graph Algorithms And Applications
The ''Journal of Graph Algorithms and Applications'' is an open access peer-reviewed scientific journal covering the subject of graph algorithms and graph drawing. The journal was established in 1997 and the editor-in-chief is Giuseppe Liotta (University of Perugia). It is abstracted and indexed by Scopus and MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links .... MathSciNet, retrieved 2011-03-02. References External links *{{Official website, http://jgaa.info/ ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
