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The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K 5 nor the complete bipartite graph K 3,3. [1] The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions ...
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 ...
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors' positions.
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.
The British 8-bit BBC Micro has partial elements of a planar pixel arrangement. The Slovak PP 01 includes a 24KB plane-based 8-colour graphics mode with a resolution of 256x256 pixels. The 16-bit Atari ST and Amiga platforms from the 80s and 90s were exclusively based on a planar graphics configuration alongside a powerful blitter .
Since there is no way of partitioning the faces into two subsets that produce a sum obeying Grinberg's theorem, there can be no Hamiltonian cycle. [1] For instance, for the graph in the figure, all the bounded faces have 5 or 8 sides, but the unbounded face has 9 sides, so it satisfies this condition on numbers of sides and is not Hamiltonian.
The other, more difficult, direction of Steinitz's theorem states that every planar 3-connected graph is the graph of a convex polyhedron. There are three standard approaches for this part: proofs by induction, lifting two-dimensional Tutte embeddings into three dimensions using the Maxwell–Cremona correspondence, and methods using the circle ...
Planar was founded on May 23, 1983 [3] by Jim Hurd, Chris King, John Laney and others as a spin-off from the Solid State Research and Development Group of the Beaverton, Oregon, based Tektronix. [4] In 1986, a division spun off from Planar to work on projection technology and formed InFocus. [5]
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