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
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 .
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:
 : is flattenable.
 : In neither of the two Kuratowski graphs is contained as a minor .
 : In neither of the two Kuratowski graphs is included as a topological minor .
See also
literature
 Reinhard Diestel : Graph Theory (= Graduate Texts in Mathematics . Volume 173 ). 3. Edition. Springer Verlag, Berlin / Heidelberg / New York 2005, ISBN 3540261826 ( MR2159259 ).
 Dieter Jungnickel : Graphs, Networks and Algorithms . With 209 Figures and 9 Tables (= Algorithms and Computation in Mathematics . Volume 5 ). 3. Edition. Springer Verlag, Berlin / Heidelberg / New York 2008, ISBN 9783540727798 ( MR2363884 ).
 Klaus Wagner: About a property of the flat complexes . In: Mathematical Annals . tape 114 , 1937, pp. 570590 , doi : 10.1007 / BF01594196 ( MR1513158 ).
 Klaus Wagner: Theory of graphs (= BI university pocket books. 248 / 248a). Bibliographisches Institut, Mannheim ( inter alia ) 1970, ISBN 3411002484 ( MR0282850 ).