Nonplanar

Nonplanar describes the property of an object that cannot be represented in a two-dimensional plane without overlaps or intersections. In mathematics, the term is most commonly applied to graphs, though it can also describe molecular geometry in chemistry.

In graph theory, a graph is nonplanar if it cannot be drawn on the plane so that its edges intersect only at their endpoints. A drawing with no edge crossings is called a planar embedding. The classic examples of nonplanar graphs are the complete graph on five vertices, K5, and the complete bipartite graph K3,3. Kuratowski’s theorem characterizes nonplanarity: a graph is nonplanar if and only if it contains a subdivision of K5 or K3,3 as a subgraph. This result leads to the study of graph minors and related planarity criteria, including various planarity testing algorithms.

Planarity testing asks whether a given graph can be drawn without edge crossings, and if so, to produce a planar embedding. Algorithms exist with linear-time performance, such as those developed by Hopcroft and Tarjan, and others based on improved minor testing.

In chemistry, molecules are termed nonplanar when their atoms cannot all lie in the same geometric plane. Nonplanarity in molecules often arises from sp3-hybridized centers or steric hindrance, and it influences properties such as chirality and reactivity. Examples include many cyclic or bridged compounds and certain organometallic structures.

Overall, nonplanarity highlights inherent spatial constraints that prevent a simple two-dimensional representation, contrasting with the notion of planar objects or systems that can be laid out on a plane without crossings.