# Wagner theorem

The set of Wagner , english Wagner's theorem is a proposition from the mathematical branch of topological graph theory , which in 1937 by the mathematician Klaus Wagner published was. The theorem is related to Kuratowski's theorem and, like the latter, gives a characterization of flattenable graphs .

## Formulation of the sentence

Fig. 1: The Kuratowski graph ${\ displaystyle K_ {5}}$
Fig. 2: The Kuratowski graph ${\ displaystyle K_ {3,3}}$

The sentence is as follows:

A finite, simple graph can be flattened if and only if it does not contain a subgraph that is contractible to one of the two Kuratowski graphs and .${\ displaystyle G = (V, E)}$ ${\ displaystyle K_ {5}}$${\ displaystyle K_ {3,3}}$

## application

With Wagner's theorem it can be shown that the Petersen graph cannot be flattened.

## Inference

The two sentences by Kuratowski and Wagner together lead to the following result:

For a finite simple graph, the following are equivalent:${\ displaystyle G = (V, E)}$
${\ displaystyle (I)}$  : is flattenable.${\ displaystyle G}$
${\ displaystyle (II)}$  : In neither of the two Kuratowski graphs is contained as a minor .${\ displaystyle G}$
${\ displaystyle (III)}$  : In neither of the two Kuratowski graphs is included as a topological minor .${\ displaystyle G}$