Graph Theory/Planar Graphs

Planar Graphs

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Definition

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

Characterization

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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.