Plane Drawing

graph-theory

Definition

Plane Drawing

A plane drawing of a graph $G=(V,E)$ is a placement of $G$ in the plane ${\mathbb{R}}^2$ such that each vertex is drawn as a distinct point and each edge is drawn as a curve between its endpoints, with no two edges crossing except at a shared endpoint.

Distinction

A planar graph is an abstract graph that has at least one plane drawing. A plane graph is the graph together with one fixed plane drawing.

Thus planarity is a property of the abstract graph, while a plane drawing is a particular representation in ${\mathbb{R}}^2$.

Edges

Edges in a plane drawing may be represented by curves. By Fáry’s theorem, this does not change the class of planar graphs: every graph with a crossing-free curved drawing also has a crossing-free straight-line drawing.