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Topic: Dual graph

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	www.jpowered.com /php-scripts/horizontal-bar-graph (298 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane.
		A graph is called outerplanar if it has an embedding in the plane such that the vertices lie on a fixed circle and the edges lie inside the disk of the circle and don't intersect.
		Duals are useful because many properties of the dual graph are related in simple ways to properties of the original graph, enabling results to be proven about graphs by examining their dual graphs.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=planar_graph (1404 words)

	dual (Site not responding. Last check: )
		Definition: The dual of a planar graph, G, is a graph with a vertex for each region in G and an edge between vertices for each pair of adjacent regions.
		The original graph is the dual of the dual.
		Herman Servatius' Self-dual maps, that is, planar graphs that are duals of themselves.
	www.nist.gov /dads/HTML/dual.html (166 words)

	NationMaster - Encyclopedia: Dual graph (Site not responding. Last check: )
		Dual graph is a term used in the mathematical study of graphs.
		Dual graphs are not unique, in the sense that a same graph can have non-isomorphic dual graphs (since the dual graph depends on a particular plane embedding).
		A geometric graph is a graph drawn in the plane with straight line segments as edges connecting vertices that are assumed to be in general position.
	www.nationmaster.com /encyclopedia/Dual-graph (693 words)

	Dual polyhedron - Biocrawler (Site not responding. Last check: )
		The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges.
		The vertices of the dual, then, are the reciprocals of the face planes of the original, and the faces of the dual lie in the reciprocals of the vertices of the original.
		It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph.
	www.biocrawler.com /encyclopedia/Dual_polyhedron (573 words)

	Planar graph
		In graph theory, a planar graph is a graph that can be drawn on a piece of paper so that no edges intersect.
		If, given the graphs A and B, and B which is an expansion of A, it is often described that A is homeomorphic to B. In practice, Kuratowski's criterion cannot be used to quickly decide whether a given graph is planar.
		For a planar graph G we may construct a graph whose vertices are the regions into which G divides the plane (including a single external region).
	www.ebroadcast.com.au /lookup/encyclopedia/pl/Planar_graph.html (417 words)

	Colorful Mathematics: Part I
		The number of edges in a graph and its dual graph are the same.
		Note that when the original graph has only 3-valent vertices, the dual graph has faces which are all 3-sided (3-gon or triangles).
		The graph above has a clique with 4 vertices which is not a subgraph of a clique with 5 vertices, and a clique with 5 vertices.
	www.ams.org /featurecolumn/archive/coloring4.html (665 words)

	Sparse Matrix Ordering Example
		Graph theory was identified as a powerful tool for sparse matrix computation when Seymour Parter used undirected graphs to model symmetric Gaussian elimination more than 30 years ago [28].
		Graphs can be used to model symmetric matrices, factorizations and algorithms on non-symmetric matrices, such as fill paths in Gaussian elimination, strongly connected components in matrix irreducibility, bipartite matching, and alternating paths in linear dependence and structural singularity.
		By using a dual graph of a mesh, one way to implement the algorithm family of constructing a PSAW is to reuse BGL algorithms such as BFS and DFS with a customized visitor to provide operations during traversal.
	www.boost.org /libs/graph/doc/sparse_matrix_ordering.html (1346 words)

	Planar graph (Site not responding. Last check: )
		In graph theory, a planar graph is a graph that can be embedded in a plane so that no edgess intersect.
		Every planar graph without loops is 4-partite, or 4-colorable; this is the graph-theoretical formulation of the four color theorem.
		A graph is called outerplanar if it has an embedding in the plane such that the vertices lie on a fixed circle and the edges lie inside the disk of the circle and don't intersect.
	www.xasa.com /wiki/en/wikipedia/p/pl/planar_graph.html (1105 words)

	CaGe -- Definitions
		Dualizing exchanges vertex degree and face size: The degree of each dual vertex is clearly equal to the size of the corresponding original face: each edge bounding the original face creates a dual edge starting at the dual vertex.
		Likewise, a dual face's size is equal to the degree of the corresponding original vertex: observe how the dual graph's faces form around the original vertices, bounded by those dual edges that cross the original edges starting at the original vertex.
		An atom in chemistry is represented by a vertex in graph theory.
	www.mathematik.uni-bielefeld.de /~CaGe/definitions.html (988 words)

	Lectures and Colloquia (Site not responding. Last check: )
		For graphs on more general (orientable) surfaces, a coloring still implies existence of a nowhere-zero flow in the dual graph, but the converse is far from being true.
		This is a recent result that involves the notion of circular chromatic number and circular flows which generalize the corresponding classical parameters.
		They define this polynomial in a recursive way, that is, the interlace polynomial of a graph is the sum of the interlace polynomials of two smaller graphs.
	www.math.tu-berlin.de /MDS/cgc/Veranstaltungen/Vorlesungen/WS01/vl-17.12.html (222 words)

	Article
		The dual is constructed by designating a capital point in each region and then defining line segments between the capital points of adjacent regions.
		In this case, the dual graph reflects the presence of 1-sided regions by the lack of closed circuits in the graph, making it more obvious that 2 colors suffice for properly coloring such map-outlines.
		Separation of points A and B in the graph requires 4 line segments giving passage around the C-D line segment by way of the C and D regions, as if edges had no length and regions were nothing but adjacent to each other as in Figure 1.
	www.sas.org /tcs/weeklyIssues/2004-05-28/feature2/index.html (2010 words)

	NationMaster - Encyclopedia: Connectivity (Site not responding. Last check: )
		The term connectivity is also used in Graph theory, where it extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency.
		One major problem that has plagued graph theory since its inception is the lack of consistency in terminology.
		Type A USB connector Dual images of the two Type B USB connectors, mini and full size, side and front view, compared with a U.S. 5Â¢ piece (nickel) in both images for scale.
	www.nationmaster.com /encyclopedia/Connectivity (885 words)

	Hamiltonian Cycles in the Vertex-Adjacency Dual
		The dual graph of a triangulation is obtained by defining a vertex for each triangle and drawing an edge between two traingles which share an edge (figure 3).
		Fig 5 : A sequential triangulation and its dual graph
		In [6], Chen, Grigni and Papadimitriou define the map graph of a planar subdivision P (or a map) to be a graph G where the vertices of G correspond to the faces of P and two vertices u and v are adjacent if their corresponding faces in P share any point on their boundary.
	cgm.cs.mcgill.ca /~athens/cs507/Projects/2004/Perouz-Taslakian (1100 words)

	Coordinate bisection
		87] partitions the graph according to the coordinates of the vertices.
		For this purpose the coordinate of a vertex of the communication graph is taken as that of its corresponding node of the mesh, or that of the centroid of the corresponding element in the case of a dual graph.
		Figure 3 shows the result of applying the RCB to the dual graph of a mesh with 788 elements.
	www.dl.ac.uk /TCSC/Staff/Hu_Y_F/PROJECT/pdcp_siam/node5.html (190 words)

	Introduction (Site not responding. Last check: )
		Both resolution graphs and splice diagrams are labelled graph-like diagrams used to encode geometric data closely related to some resolution of singularities procedure in algebraic geometry.
		Instead they work by having a directed graph, referred to as the underlying graph, as a primary attribute and by caching other data (which is typically associated to particular vertices and edges of the graph) in sequences as secondary attributes.
		Graph surgery routines cannot be used directly since they must manage both the underlying graph and the associated data.
	www.math.niu.edu /help/math/magmahelp/text992.html (319 words)

	Graphs for FreeWX and FreeWX-Wi - V2 Introduction
		The first graph below is a standard dual graph using a calendar based X axis.
		This next graph is a MinMax Graph with a zero based X axis (and smaller).
		The graph below is the same graph but this time using a generic 1 to 30 X axis and with the legend removed.
	www.dandmbr.co.uk /jsgraphit/v2Intro.htm (314 words)

	Triangle Strip Tunneling Algorithm			The dual graph of a mesh has a node corresponding to each face of the mesh and has an edge between two nodes whose corresponding faces are adjacent.
		Each edge of the dual graph is either a "strip edge" or a "nonstrip edge".
		A tunnel in the dual graph is a sequence of edges that joins the ends of two strips.
	www.cs.queensu.ca /home/jstewart/strips/algorithm (966 words)

	Planar graph - Definition, explanation
		A finite graph is planar if and only if it does not contain a subgraph that is an expansion of K
		A finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to K
		PIGALE is a graph editor and an algorithm library essentially concerned with planar graphs.
	www.calsky.com /lexikon/en/txt/p/pl/planar_graph.php (1364 words)

	Quadrilateral and Tetrahedral Mesh Stripification Using the 2-factor of the Dual Graph — UC Irvine - Computer ...
		In this paper, we propose algorithms that find a 2-factor of a graph, if one exists, for a restricted class of graphs in which all vertices have degree four or less, in O(n2) complexity where n is the number of vertices of the graph.
		Such graphs are actually dual graphs of quadrilateral and tetrahedral meshes that are widely used in graphics and visualization applications.
		We use the similarity between the dual graphs of the quadrilateral and tetrahedral meshes to introduce a novel, unified graph based algorithm to produce quad and tetrahedral strip representations respectively.
	www.graphics.ics.uci.edu /publications/DG05 (304 words)

	[H] Enthusiast - ASUS A8R32-MVP Deluxe
		This is quite frankly a terrible bar graph to show you, but some of the changes in the last few weeks forced our hand on this.
		This is an interesting graph here, as DivX Create 6.1 has been widely trumpeted as one of the few dual core mainstream solutions on the market.
		Video editing software is where dual core processors really start showing their value when testing for speed and seeing real-world benefits, outside of creamy smooth multitasking.
	www.hardocp.com /article.html?art=OTY5LDYsLGhlbnRodXNpYXN0 (629 words)

	War Story: Getting the Graph (Site not responding. Last check: )
		The task of finding a small number of strips that cover each triangle in a mesh could be modeled as a graph problem, where the graph has a vertex for every triangle of the mesh, and there is an edge between every pair of vertices representing adjacent triangles.
		My dual graph is going to have as many vertices as the number of triangles.
		The next day she reported that the graph could be built in seconds, even for much larger models.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK2/NODE62.HTM (826 words)

	College of Physical and Mathematical Sciences : Spring Research Presentation (Site not responding. Last check: )
		When graphing multivariable polynomials in tropical space, every polynomial can be associated with a dual graph, which can help to calculate the graph of a function.
		These dual graphs also restrict the choices of graphs for these multivariable polynomials.
		Furthermore, there is a simple algorithm that allows for the creation of the dual graph of a conic polynomial, almost by inspection.
	cpms.byu.edu /springresearch/abstractentry.php?id=470 (130 words)

	graph theory: duality between indepdent set and graph matching?
		Dear all, I try to establish duality between independent set and graph matching.
		My confusion comes from that if you map all edges to in graph G to vertices in dual graph G', and map all shared vertices in G to edges in G', the dual graph G' loses edges.
		The vertices with odd-degree in G cannot be mapped to be edges in graph G'.
	www.forum-one.org /new-6147048-4346.html (123 words)

	Graph Theory Lesson 22 (Site not responding. Last check: )
		The swap menu item is on the small screen where you have both the graph and its dual displayed.
		The program tries to put the points of the dual graph in positions that correspond to the center of the region they represent.
		Actually, the dual of a platonic graph is always a platonic graph.
	oneweb.utc.edu /~Christopher-Mawata/petersen/lesson22.htm (370 words)

	Math Forum Discussions
		The condition on the dual graph appears to be difficult to visualize,
		the dual graph, this translates to the condition that the group of
		correct that this implies the even-cycle condition on the dual graph),
	www.mathforum.org /kb/thread.jspa?messageID=3686871&tstart=0 (485 words)

	Graphs in Image Analysis - Introduction
		Dual graph contraction reduces image graphs while preserving important structural properties.
		The dual graph contraction animation shows the scheme of the dual graph contraction and allows the user to observe each step.
		Dual graph contraction is used, among other things, for connected component analysis and segmentation.
	www.prip.tuwien.ac.at /Research/Graphs/dgc (66 words)

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