# Graph Theory/Planar Graphs

# Planar Graphs

edit### Definition

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

### Characterization

editThe 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 finite graph is planar if and only if it does not contain a subgraph that is a subdivision of or .

A subdivision of a graph results from inserting vertices into edges zero or more times.

Instead of considering subdivisions, Wagner's theorem deals with minors:

- A finite graph is planar if and only if it does not have or as a minor.

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.