Graph Theory/Planar Graphs

= Planar Graphs =

Definition
A planar graph is a graph that can be drawn in the plane such that there are no edge crossings.

Characterization
The planar graphs can be characterized by a theorem first proven by the Polish mathematician Kazimierz Kuratowski in 1930, now known as Kuratowski's theorem: A subdivision of a graph results from inserting vertices into edges zero or more times.
 * A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of $$K_5$$ or $$K_{3,3}$$.

Instead of considering subdivisions, Wagner's theorem deals with minors: A graph H is a minor of a graph G if a copy of H can be obtained from G via repeated edge deletion and/or edge contraction.
 * A finite graph is planar if and only if it does not have $$K_5$$ or $$K_{3,3}$$ as a minor.