A planar network (or graph) is one that can be drawn on a plane without having any edges cross.

For these graphs we can define the Dual Graph, with vertices being faces (regions completely enclosed by edges), and edges being among faces that share an edge of the original graph. This new graph is also planar

Dual graphs were used to prove the four-color theorem by Appel and Haken, which translated to graphs is stated in terms of the chromatic number, the number of colors required to color the vertices of a graph in such a way that no two vertices connected by an edge have the same color.

Kuratowski's theorem...

As of yet, there is no popular measure of degree of planarity (i.e. how planar a graph is?)