Nichtplanare

Nichtplanare is a term used in geometry and graph theory to describe objects that cannot be embedded in a plane without crossings. In common mathematical usage, an object is non-planar if no drawing on the plane can realize its connections without intersecting edges.

In graph theory, planarity distinguishes graphs that can be drawn on the plane without edge crossings from those that cannot. A planar graph can be drawn in the plane so that its edges are curves meeting only at their endpoints. Non-planar graphs require at least one crossing in every plane drawing.

Kuratowski's theorem provides a practical criterion: a finite graph is non-planar if and only if it contains a subdivision of K5 (the complete graph on five vertices) or K3,3 (the complete bipartite graph with partitions of three vertices). Wagner's theorem offers an equivalent perspective: a graph is non-planar if it has K5 or K3,3 as a minor.

Examples of non-planar graphs include K5 and K3,3; these are the classic forbidden subgraphs for planarity. Some non-planar graphs can be embedded without crossings on surfaces of higher genus, such as a torus, but not on the plane.

Planarity can be tested efficiently; modern algorithms run in linear time, such as the Hopcroft–Tarjan planarity test. In other disciplines, the term non-planar also describes objects that are not flat, such as certain molecular conformations in chemistry.