nonplanar

Nonplanar sections require 3D analysis and are a research area.

However, nonplanar triangle-free graphs may require many more than three colors.

It is the only known nonplanar cubic partial cube.

In indigo white, the conjugation is interrupted because the molecule is nonplanar.

Every Möbius ladder is a nonplanar apex graph.

The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.

And conversely, every nonplanar linkless graph has multiple linkless embeddings.

The Petersen graph is nonplanar.

However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings.

The primary challenges to integrating nonplanar multigate devices into conventional semiconductor manufacturing processes include:

Kazimierz Kuratowski proved in 1930 that K is nonplanar, and thus that the problem has no solution.

When H is nonplanar, we also need to consider k-clique-sums of a list of graphs, each of which is embedded on a surface.

As a Möbius ladder, the Wagner graph is nonplanar but has crossing number one, making it an apex graph.

This conjecture is a strengthened form of the four color theorem, because any graph containing the Petersen graph as a minor must be nonplanar.

A Kuratowski subgraph of a nonplanar graph can be found in linear time, as measured by the size of the input graph.

For instance, in the minor-minimal nonplanar graphs K and K, any of the vertices can be chosen as the apex.

Nonplanar devices are also more compact than conventional planar transistors, enabling higher transistor density which translates to smaller overall microelectronics.

Focus stacking also allows generation of images physically impossible with normal imaging equipment; images with nonplanar focus regions can be generated.

The two graphs K and K are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula.

The more difficult direction in proving Kuratowski's theorem is to show that, if a graph is nonplanar, it must contain a Kuratowski subgraph.

Nonplanar graphs may also be parameterized by their crossing number, the minimum number of pairs of edges that cross in any drawing of the graph.

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar faces or vertex figures.

A regular skew polygon in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform antiprism.

The theorem cannot be generalized to all nonplanar triangle-free graphs: not every nonplanar triangle-free graph is 3-colorable.

If a polyhedron is not simple (it has more than three edges at a vertex) the line graph will be nonplanar, with a clique replacing each high-degree vertex.