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Kuratowski's theorem
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==Kuratowski subgraphs== {{tesseract_graph_nonplanar_visual_proof.svg}} 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 = 743β767 | 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.
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