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Topic: Planar dual

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	Planar graph - Wikipedia, the free encyclopedia
		All finite or countably infinite trees are outerplanar and hence planar.
		Every planar graph without loops is 4-partite, or 4-colorable; this is the graph-theoretical formulation of the four color theorem.
		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.
	en.wikipedia.org /wiki/Planar_graph (1532 words)

	Planar graph
		Euler's formula states that if a connected planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer region), then
		Every planar graph is 4-partite, or 4-colorable; this is the graph-theoretical formulation of the four color theorem.
		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)

	[No title]
		In fact, it is not hard to see that the dual of the dual of an imbedded graph is the original graph.
		The dual of a triangulation is an imbedded cubic (trivalent) graph.
		The dual is a 1-connected planar cubic graph, possibly with double edges but with no faces of size less than 3.
	cs.anu.edu.au /people/Brendan.McKay/plantri/plantri-guide.txt (3805 words)

	Logical graph - Wikipedia, the free encyclopedia
		There are two types of duality that have to be kept separately mind in the use of logical graphs, logical duality and topological duality.
		The planar regions of the original graph correspond to nodes (or points) of the dual graph, and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph.
		For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as "()() = ()".
	en.wikipedia.org /wiki/Logical_graph (8875 words)

	dual (Site not responding. Last check: 2007-10-30)
		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 (177 words)

	A Self-Dual Hendecahedron
		The principle of duality states that for every polygon there is a dual, or reciprocal, polygon, whose edges correspond to the vertices (corner points) of the original and whose vertices likewise correspond to its edges.
		To obtain the dual of a polygon, the original is reciprocated with respect to a circle.
		To obtain the dual of a polyhedron, the original is reciprocated with respect to a sphere.
	www.steelpillow.com /polyhedra/selfdualhen/selfdualhen.htm (996 words)

	A Survey of Venn Diagrams: Graphs Associated with Venn Diagrams
		For example, the Venn dual of the three circle Venn diagram is a planar embedding of the 3-cube, Q
		A theorem of Whitney [Wh] states that a plane embedding of a 3-connected planar graph is unique, once the outer face has been identified.
		The directed Venn dual is a spanning subgraph of the Hasse diagram (regarded as a directed graph) of the Boolean lattice.
	www.theory.cs.uvic.ca /~cos/venn/VennGraphEJC.html (2622 words)

	Schnyder's Grid-Embedding Algorithm
		The input to Schnyder's algorithm is assumed to be a planar graph, without any self-loops or multiple adjacencies, with the topology of a planar layout already specified in terms of a clockwise ordering of the edges at each vertex, as results from Hopcroft-Tarjan or other linear-time planar embedding algorithms.
		Proof: As we saw when discussing properties of planar graphs, the weak dual of G (the graph formed from the planar dual by removing the dual of f) is a tree.
		Conversely, any planar triangulation has a vertex along this path that can be contracted into the outer corner without creating multiple adjacencies (by Lemma 1, applied to the vertices of the path), and one can construct a realizer of the original graph by uncontracting a realizer of the contracted graph.
	www.ics.uci.edu /~eppstein/gina/schnyder (1091 words)

	THE DETECTION OF METASTATIC PROSTATE CANCER WITH SIMULTANEOUS DUAL RADIOISOTOPE SPECT IMAGES
		If significant colonic activity was seen on planar images on day 5, the imaging was discontinued and the patient was asked to take another dose of cathartic and return for imaging on day 6.
		During the RBC labeling procedure, whole-body images and planar images of the chest, abdomen, and pelvis were obtained.
		Whole-body planar imaging was found to be essential, as a left supraclavicular node in one patient and single lung lesion in two patients would not have been found with SPECT imaging of the abdomen and pelvis alone.
	www.uic.edu /com/uhrd/nucmed/rsnaej/prost_3.htm (7027 words)

	High performance circularly polarized microstrip antenna (US5661494)
		A microstrip antenna for radiating circularly polarized electromagnetic waves comprising a cluster array (20) of at least four microstrip radiator elements (22a-22d), each of which is provided with dual orthogonal coplanar feeds in phase quadrature relation achieved by connection to an asymmetric T-junction power divider (30) impedance notched at resonance.
		The dual fed circularly polarized reference element is positioned with its axis at a 45° angle with respect to the unit cell axis.
		The other three dual fed elements in the unit cell are positioned and fed with a coplanar feed structure with sequential rotation and phasing to enhance the axial ratio and impedance matching performance over a wide bandwidth.
	www.delphion.com /details?pn=US05661494__ (437 words)

	Tom's Combinatorial Geometry Class
		A graph is called a planar graph if it is drawn in such a way that the edges never cross, except at where they meet at vertices.
		There is a direct connection between polyhedra and planar graphs, namely that we can take any polyhedron and "project" it down onto a flat piece of paper, turning it into a graph.
		And since the same must be true of the dual of this Platonic solid, we have k < = 5.
	www.merrimack.edu /~thull/combgeom/graphnotes.html (1055 words)

	Euler's Formula
		For any connected embedded planar graph G define the dual graph G* by drawing a vertex in the middle of each face of G, and connecting the vertices from two adjacent faces by a curve e* through their shared edge e.
		It fits in well with other topics of planar duality such as the fact that every planar graph with all faces even is bipartite (by duality from Euler tours).
		This argument is the planar dual to the proof by induction on faces.
	www.ics.uci.edu /~eppstein/junkyard/euler/all.html (5087 words)

	Hyperbolic Planar Tesselations
		The Omnitruncated {3,7} is the "most nearly planar" of all semiregular or regular hyperbolic tesselations, in the sense that if you tried to construct it from Euclidean planar polygons, the sum of the angles at each vertex would be as small as possible while exceeding 360 degrees.
		We'll consider spherical, planar, and hyperbolic tilings all at once, using "Schwarz polygons" (a generalization of Schwarz triangles) to generate the symmetry groups, and using a generalized Coxeter-Dynkin symbol to name the resulting tesselations.
		We can color each vertex of the uniform tiling (or Schwarz polygon of the dual) "even" or "odd" depending on whether it is generated as an even or odd number of reflections of a fixed initial vertex (or Schwarz polygon of the dual, respectively), i.e.
	www.plunk.org /~hatch/HyperbolicTesselations (1397 words)

	Planar Handbook Excerpt
		The dharculus is a creature from an alternate reality far from the Material Plane, and perhaps beyond the planar cosmology as it is presently understood.
		It may be that these creatures are dependent on some element from their unknown place of origin; however, reports indicate that more and more of these creatures are hunting along the shoals of corporeal planes.
		The maw and main body of the creature remain only a threatening thickness in the air even when the bite is delivered, so the sudden jerking and wounding of a victim become all the more unsettling to the victim's compatriots.
	www.wizards.com /default.asp?x=dnd/ex/20040706a&page=5 (853 words)

	Graphs and Digraphs (Site not responding. Last check: 2007-10-30)
		The planarity test is based on a variant of the Hopcroft-Tarjan planarity algorithm.
		When a graph is found to be planar, its planar dual is automatically constructed.
		The planar layout algorithm is that of W. Kocay and C. Pantel, "An algorithm for finding a planar layout of a graph with a regular polygon as outer face", Utilitas Mathematica 48 (1995), 161-178.
	kohlrabi.cs.umanitoba.ca /G&G/GraphOverview.html (1016 words)

	Product Details
		The 19” Planar PX1910M dual input LCD monitor features Planar’s RapidVideo™ accelerator for superb full-motion video performance at an incredibly fast 8 ms response time.
		Planar Systems' PX212M is a good 21-inch unit that easily rotates between horizontal and vertical orientations.
		With "one of the industry's fastest response times (16ms)", the Planar PX171M was rated a top pick by PC World at Comdex '02...
	www.planar.com /products/flatpanel_monitors/px_flatpanel/Print_layout_1_11831_11831.cfm?comparator=2113 (621 words)

	Planar dual polarization antenna (US4903033)
		A microwave-frequency microstrip antenna (10) simultaneously usable for both transmitting and receiving microwave-frequency signals that have dual orthogonally polarized components.
		A second dielectric (22) abuts the ground plane (20) and has embedded thereon two substantially identical conductive planar feed networks (52,51) that are disposed at right angles to each other.
		At least one additional optional dielectric layer (16,18) having a conductive patch (36,34) may be interposed between the first dielectric (12) and the ground plane (20) for purposes of broadening the bandwidth of the antenna (10).
	www.delphion.com /details?pn10=US04903033 (443 words)

	Exploring the repertoire of RNA secondary motifs using graph theory; implications for RNA design -- Gan et al. 31 (11): ...
		The ambiguities in representing the topology of RNA secondary structures using dual graphs are resolved by employing digraphs; however, the topologies of single-stem structures (rows 1 and 2 of column E) cannot be differentiated by both dual and digraph representations.
		A planar graph is a set of nodes or vertices (•) and a set of line segments or edges (—) in a plane where each of the segments either joins two vertices or joins a vertex to itself.
		In contrast, in dual graphs, the unmatched base pairs are considered as edges and the vertices are helical stems.
	nar.oxfordjournals.org /cgi/content/full/31/11/2926 (8835 words)

	Dictionary of Combinatorics -- E (Site not responding. Last check: 2007-10-30)
		Planar bipartite graphs are dual to planar Eulerian graphs and vice versa.
		A formula relating the number of edges, and nodes of a graph to the number of regions into which it will divide the plane.
		Let G be a connected planar graph with n nodes and e edges which divides the plane into r regions.
	www.southernct.edu /~fields/comb_dic/E.html (203 words)

	Planar : ComputerHQ
		The Planar 17" PL1700 is ideal for small and medium corporate business leaders and the applications that make their businesses succeed.
		The Planar PL1700M is an analog 17-inch TFT-LCD monitor delivering a 450:1contrast ratio, 260 nits brightness, 1024 x 768 recommended resolution, a horizontal/vertical viewing angle of 150/135...
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	www.computerhq.com /ipod/parts-brand-planar-page-2.html (533 words)

	THE DETECTION OF METASTATIC PROSTATE CANCER WITH SIMULTANEOUS DUAL RADIOISOTOPE SPECT IMAGES
		If significant colonic activity was seen on planar images on day 5, the imaging was discontinued and the patient was asked to take another dose of cathartic and return for imaging on day 6 post infusion.
		The overall effect of the new dual isotope SPECT imaging protocol in the 20 study patients was the clinical upstaging of 3 patients (previously thought to be disease free).
		The overall clinical outcome of the new dual isotope SPECT imaging protocol was the upstaging of 3 patients (previously thought to be disease free).
	www.uic.edu /com/uhrd/nucmed/rsnaej/prostate.htm (7024 words)

	Math3343 Assignment 5, Fall 2002 (Site not responding. Last check: 2007-10-30)
		Let G be a self-dual graph, with f faces, m edges and n vertices.
		G's dual graph must have f vertices and n faces, since vertices become faces and faces become vertices in the dual graph.
		Since the dual graph is isomorphic to G, it must have the same number of vertices as G, so
	www.cs.unb.ca /profs/horton/math3343/solutions5.html (671 words)

	Miniaturized HTS/Dielectric Multilayer Filters for Satellite Communications
		A major limiting factor in the continuing improvement of satellite communication systems is the unavailability of high-performance (i.e., high-Q) miniaturized filters that are compatible with monolithic microwave integrated circuits (MMIC) components.
		Coupling between the dual orthogonal modes supported by these resonators is achieved by introducing a perturbation to the symmetry of the previously single-mode resonator at a location that is offset 45° from the axes of coupling to and from the resonator.
		Proof-of-concept of these miniaturized multilayer filters has already been demonstrated in our laboratory (ref. 3), and work is underway to optimize their performance through more detailed analysis, fabrication, and testing.
	www.grc.nasa.gov /WWW/RT1996/5000/5620m2.htm (503 words)

	A Survey of Venn Diagrams: Open Problems
		Equivalently: Is every planar dual graph of a simple Venn diagram Hamiltonian?
		The dual of a simple Venn diagram is a maximal planar spanning subgraph of the hypercube.
		(The dual is still maximal if the Venn diagram is only rigid.) Does every maximal planar 3-connected spanning subgraph of the hypercube, where all faces are quadrilaterals, occur as the dual graph of some simple Venn diagram?
	sue.csc.uvic.ca /~cos/venn/VennOpenEJC.html (858 words)

	VR-Zone : Technology Beats - Infineon 2GB DDR2 Planar Registered DIMM
		The 2 GB DDR2 Planar Registered DIMM, which is expected to become the preferred density of high-end server suppliers, targets high-performance data processing and storage applications in the server and storage infrastructure market.
		With the introduction of the 2 GB DDR2 Planar Registered DIMM, Infineon optimizes its DDR2 standard portfolio of registered modules with memory densities of 256 MB, 512 MB, 1 GB, 2 GB and 4 GB.
		Samples of the 2 GB DDR2 Planar Registered DIMM are available in an organization of 256 Mbit x 72 (2 Rank x4), for speeds of PC2-3200 and PC2-4300 at unit prices from US-Dollar 700,- to 910,-.
	www.vr-zone.com /?i=691&s=1 (443 words)

	Survey of Venn Diagrams: Graphs
		With each Venn diagram, C, we may associate another plane graph called the Venn graph, and denoted G(C), which is the planar dual of the Venn diagram.
		It's vertices are the connected open regions (faces) from the definition of Venn diagrams.
		The proof of [CHP96] makes use of the radual graph, R(C) of the Venn diagram, which, for an arbitrary plane graph, is the union of the radial graph and the dual graph (see Ore [Or]).
	www.maths.tcd.ie /EMIS/journals/EJC/Surveys/ds5/VennGraphEJC.html (2303 words)

	74.434 Assignment 2 (Site not responding. Last check: 2007-10-30)
		Please ensure that your name and student number appear on the assignment, which must be submitted in a folder.
		A planar graph is self-dual if it is isomorphic to its dual.
		Show that a planar graph is bipartite iff its dual is eulerian.
	www.cs.umanitoba.ca /~cs434/A2_434.html (494 words)

	CURRICULUM VITAE (Site not responding. Last check: 2007-10-30)
		Pinontoan and Richter [PR] described a method of computing the crossing numbers of crossing-critical graphs constructed by gluing planar tiles in a circular way (wiht special gluing applied to one of the tiles), namely by computing the crossing number of some tiles arranged in a linear way.
		A rectangular dual G* of an n-vertex graph G, is comprised of n non-overlapping rectangles with the following properties: (i) each vertex v of G corresponds to a distinct rectangle v* in the rectangular dual and (ii) if uv is an edge in G, then rectangles u* and v* are adjacent in the rectangular dual.
		A properly triangulated planar (PTP) graph is a connected planar graph with the following properties: (i) every face (except the exterior) is a triangle (i.e.
	www.minahasaraya.net /benny (1302 words)

	A Survey of Venn Diagrams: Symmetric Diagrams
		Think of the diagram as being projected stereographically onto a sphere with the axis orthogonal to the plane running through the point of symmetry, and then project the diagram back onto the plane from the antipodal point on the axis.
		If the corresponding Venn dual graphs are isomorphic as plane graphs, then the diagram has polar symmetry.
		Briefly, the idea is to create the dual graph of one sector of the diagram by building a series of chains that span a set of strings from the Boolean lattice, with the property that when this dual graph is rotated, the dual of the entire Venn diagram is created.
	sue.csc.uvic.ca /~cos/venn/VennSymmEJC.html (1903 words)

	DatasheetArchive: Semiconductor Datasheets Page 1157 (Site not responding. Last check: 2007-10-30)
		Dual NPN Planar Transistors in a Hermetically Sealed Ceramic Surface Mount Package for High Reliability Applications
		Dual Bipolar NPN Devices in a Hermetically Sealed LCC2 Ceramic Surface Mount Package for High Reliability Applications
		Dual NPN Transistors in a Hermetically Sealed Ceramic Surface Mount Package for High Reliability Applications
	www.datasheetarchive.com /datasheet/pdf/1157.html (190 words)

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