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 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.