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Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. Wagner published both theorems in 1937, [1] subsequent to the 1930 publication of Kuratowski's theorem, [2] according to which a graph is planar if and only if it does not contain as a subgraph a subdivision of one of the same two forbidden ...
Planar Systems, Inc. is an American digital display manufacturing corporation with a facility in Hillsboro, Oregon. Founded in 1983 as a spin-off from Tektronix , it was the first U.S. manufacturer of electroluminescent (EL) digital displays.
A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A map graph is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they ...
In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II 1 subfactor. [1] They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial ), and have been used in describing the properties of Khovanov homology with respect to tangle composition.
In computer graphics, planar is the method of arranging pixel data into several bitplanes of RAM.Each bit in a bitplane is related to one pixel on the screen. Unlike packed, high color, or true color graphics, the whole dataset for an individual pixel is not in one specific location in RAM, but spread across the bitplanes that make up the display.
Coplanar waveguides play an important role in the field of solid state quantum computing, e.g. for the coupling of microwave photons to a superconducting qubit.In particular the research field of circuit quantum electrodynamics was initiated with coplanar waveguide resonators as crucial elements that allow for high field strength and thus strong coupling to a superconducting qubit by confining ...
An RF power amplifier incorporating planar circuit structures. The amplifier on the left feeds its output into a set of planar transmission line filters in the centre. The third circuit block on the right is a circulator to protect the amplifier from accidental reflections of the power back from the antenna
Steinitz's theorem states that these two conditions are both necessary and sufficient to characterize the skeletons of three-dimensional convex polyhedra: a given graph is the graph of a convex three-dimensional polyhedron, if and only if is planar and 3-vertex-connected. [5] [8]
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