# 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 finite graph is planar if and only if it does not contain a subgraph that is a subdivision of ${\displaystyle K_{5}}$ or ${\displaystyle K_{3,3}}$.

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 ${\displaystyle K_{5}}$ or ${\displaystyle K_{3,3}}$ 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.