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Universal Point Set
In graph drawing, a universal point set of order ''n'' is a set ''S'' of points in the Euclidean plane with the property that every ''n''-vertex planar graph has a straight-line drawing in which the vertices are all placed at points of ''S''. Bounds on the size of universal point sets When ''n'' ≤ 10, there exist universal point sets with exactly ''n'' points, but for all ''n'' ≥ 15 additional points are required. Several authors have shown that subsets of the integer lattice of size ''O''(''n'') × ''O''(''n'') are universal. In particular, showed that a grid of (2''n'' − 3) × (''n'' − 1) points is universal, and reduced this to a triangular subset of an (''n'' − 1) × (''n'' − 1) grid, with ''n''2/2 − ''O''(''n'') points. By modifying the method of de Fraysseix et al., found an embedding of any planar graph into a triangular subset of the grid consis ... [...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 men ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pathwidth
In graph theory, a path decomposition of a graph is, informally, a representation of as a "thickened" path graph, and the pathwidth of is a number that measures how much the path was thickened to form . More formally, a path-decomposition is a sequence of subsets of vertices of such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets,. and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness (one less than the maximum clique size in an interval supergraph of ), vertex separation number, or node searching number. Pathwidth and path-decompositions are closely analogous to treewidth and tree decompositions. They play a key role in the theory of graph minors: the families of graphs that are closed under graph minors and do not include all forests may be characterized as having bounded pathwidth, and the "vortices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIGACT News
ACM SIGACT or SIGACT is the Association for Computing Machinery Special Interest Group on Algorithms and Computation Theory, whose purpose is support of research in theoretical computer science. It was founded in 1968 by Patrick C. Fischer. Publications SIGACT publishes a quarterly print newsletter, ''SIGACT News''. Its online version, ''SIGACT News Online'', is available since 1996 for SIGACT members, with unrestricted access to some features. Conferences SIGACT sponsors or has sponsored several annual conferences. *COLT: Conference on Learning Theory, until 1999 *PODC: ACM Symposium on Principles of Distributed Computing (jointly sponsored by SIGOPS) *PODS: ACM Symposium on Principles of Database Systems (jointly sponsored by SIGAI and SIGACT) *POPL: ACM Symposium on Principles of Programming Languages *SOCG: ACM Symposium on Computational Geometry (jointly sponsored by SIGGRAPH), until 2014 *SODA: ACM/SIAM Symposium on Discrete Algorithms (jointly sponsored by the Society ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Graph Algorithms And Applications
The ''Journal of Graph Algorithms and Applications'' is a diamond open access peer-reviewed scientific journal covering the subject of graph algorithms and graph drawing. The journal was established in 1997 and the current co-editors-in-chief are Emilio Di Giacomo (University of Perugia) and Martin Nöllenburg (TU Wien). It is published by Brown University and is a member of the Free Journal Network. 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 .... MathS ... [...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 theory, 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, Pierre Rosenstiehl, and Roberto Tamassia. The first two sympo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * '' Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate, formerly the Institute for Scientific Information and Thomson Reuters. It is published online and in several different printed subject sections. History ''Current Contents ...'' Notable articles Two articles published in ''Discrete & Computational Geometry'', one by Gil Kalai in 1992 with a proof of a subexponential upper bound on the diameter of a polytope and another by Samuel Ferguson in 2006 on the Kepler conjecture on optimal three-dimensional sphere packing, earned their authors the Fulk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bend Minimization
In graph drawing styles that represent the edges of a graph by polylines (sequences of line segments connected at bends), it is desirable to minimize the number of bends per edge (sometimes called the curve complexity). or the total number of bends in a drawing.. Bend minimization is the algorithmic problem of finding a drawing that minimizes these quantities. Eliminating all bends The prototypical example of bend minimization is Fáry's theorem, which states that every planar graph can be drawn with no bends, that is, with all its edges drawn as straight line segments. Drawings of a graph in which the edges are both bendless and axis-aligned are sometimes called ''rectilinear drawings'', and are one way of constructing RAC drawings in which all crossings are at right angles. However, it is NP-complete to determine whether a planar graph has a planar rectilinear drawing, and NP-complete to determine whether an arbitrary graph has a rectilinear drawing that allows crossings.. Bend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygonal Chain
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. Variations Simple A simple polygonal chain is one in which only consecutive segments intersect and only at their endpoints. Closed A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment. A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to draw a distinction between a polygonal area and a polygonal chain. A space closed polygonal chain is also known as a skew "polygon". Monotone A polygonal chain is called ''monotone'' if there is a strai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Curve
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves include the closed convex curves (the boundaries of bounded convex sets), the smooth curves that are convex, and the strictly convex curves, which have the additional property that each supporting line passes through a unique point of the curve. Bounded convex curves have a well-defined length, which can be obtained by approximating them with polygons, or from the average length of their projections onto a line. The maximum number of grid points that can belong to a single curve is controlled by its length. The points at which a convex curve has a unique supporting line are dense within the curve, and the distance of these lines fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It only has one line of symmetry (reflection symmetry). In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half- disk, which is a two-dimensional geometric region that further includes all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex. All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle. Arithmetic and geometric means A semicircle can be used to construct th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Diagram
An arc diagram is a style of graph drawing, in which the vertices of a graph are placed along a line in the Euclidean plane and edges are drawn using semicircles or other convex curves above or below the line. These drawings are also called linear embeddings or circuit diagrams. Applications of arc diagrams include information visualization, the Farey diagram of number-theoretic connections between rational numbers, and diagrams representing RNA secondary structure in which the crossings of the diagram represent pseudoknots in the structure. Description In an arc diagram, the vertices of a graph are arranged along a line in the Euclidean plane. The edges are drawn as semicircles in one or both of the two halfplanes bounded by the line, or as smooth curves formed by sequences of semicircles. In some cases, line segments of the line itself are also allowed as edges, as long as they connect only vertices that are consecutive along the line. Variations of this drawing style in whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Book Embedding
In graph theory, a book embedding is a generalization of planar graph, planar embedding of a Graph (discrete mathematics), graph to embeddings in a ''book'', a collection of half-planes all having the same Line (geometry), line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the ''spine'', and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number. Every graph with vertices has book thickness at most \lceil n/2\rceil, and this formula gives the exact book thickness for complete graphs. The graphs with book thickness one are the outerplanar graphs. The graphs with book thickness at most two are the subhamilt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
