Graph Theory/Planar Graphs
Planar Graphs[edit | edit source]
Definition[edit | edit source]
A planar graph is a graph that can be drawn in the plane such that there are no edge crossings.
Characterization[edit | edit source]
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 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.