Editing Kuratowski's theorem (section)

==Kuratowski subgraphs==
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If <math>G</math> is a graph that contains a subgraph <math>H</math> that is a subdivision of <math>K_5</math> or <math>K_{3,3}</math>, then <math>H</math> is known as a '''Kuratowski subgraph''' of <math>G</math>.<ref>{{citation
 | last = Tutte | first = W. T. | author-link = W. T. Tutte
 | journal = Proceedings of the London Mathematical Society
 | mr = 0158387
 | pages = 743–767
 | series = Third Series
 | title = How to draw a graph
 | volume = 13
 | year = 1963
 | doi=10.1112/plms/s3-13.1.743}}.</ref>  With this notation, Kuratowski's theorem can be expressed  succinctly: a graph is planar if and only if it does not have a Kuratowski subgraph.

The two graphs <math>K_5</math> and <math>K_{3,3}</math> are nonplanar, as may be shown either by a [[Proof by cases|case analysis]] or an argument involving [[Euler characteristic|Euler's formula]]. Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph <math>G</math> has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of <math>G</math> itself. Therefore, a graph that contains a Kuratowski subgraph cannot be planar. The more difficult direction in proving Kuratowski's theorem is to show that, if a graph is nonplanar, it must contain a Kuratowski subgraph.