planeplanar

Planeplanar is not a standard term in mathematics or graph theory. In many contexts, it may be a mistaken combination of the words plane and planar, or it could be shorthand for concepts related to plane graphs or planar embeddings. The article therefore clarifies the closest established notions.

A plane graph is a graph drawn in the plane with no edge crossings, where the drawing itself is part of the object. The embedding fixes, for each vertex, the cyclic order of incident edges and thereby determines the faces of the drawing. A planar graph, by contrast, is an abstract graph that can be drawn in the plane without edge intersections in some embedding.

Planarity is a property of a graph: a graph is planar if there exists at least one plane embedding. Deciding planarity can be done efficiently; there are linear-time planarity tests. A fundamental characterization is Kuratowski’s theorem: a graph is planar if and only if it contains no subdivision of K5 or K3,3. Fáry’s theorem states that every simple planar graph can be drawn with straight-line edges, reinforcing the link between planarity and planar embeddings.

Some planar graphs have multiple distinct embeddings in the plane, while others have essentially unique embeddings up to the plane’s rigid motions. The study of plane graphs also involves dual graphs, faces, and Euler’s formula, which relates the numbers of vertices, edges, and faces.

If encountered, the term planeplanar should be interpreted from context. Most often, it is safer to assume it refers to plane graphs or planarity-related concepts rather than a distinct, widely recognized mathematical object.