Fáry's theorem

Because the re-embedding of G' was combinatorially isomorphic to G', removing from it the edges which were added to create G' leaves the face f, which is now a polygon P with at most five sides.

A similar method has been followed by Schnyder to prove enhanced bounds and a characterization of planarity based on the incidence partial order.

His work stressed the existence of a particular partition of the edges of a maximal planar graph into three trees known as a Schnyder wood.

It is so called because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.

Steinitz's theorem states that every 3-connected planar graph can be represented as the edges of a convex polyhedron in three-dimensional space.

Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers.