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Kuratowski's theorem
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==Related results== A closely related result, [[Wagner's theorem]], characterizes the planar graphs by their [[graph minor|minors]] in terms of the same two forbidden graphs <math>K_5</math> and <math>K_{3,3}</math>. Every Kuratowski subgraph is a special case of a minor of the same type, and while the reverse is not true, it is not difficult to find a Kuratowski subgraph (of one type or the other) from one of these two forbidden minors; therefore, these two theorems are equivalent.<ref>{{citation|title=Graph Theory|volume=244|series=Graduate Texts in Mathematics|first1=J. A.|last1=Bondy|author1-link=John Adrian Bondy|first2=U.S.R.|last2=Murty|author2-link=U. S. R. Murty|publisher=Springer|year=2008|isbn=9781846289699|page=269|url=https://books.google.com/books?id=HuDFMwZOwcsC&pg=PA269}}.</ref> An extension is the [[Robertson–Seymour theorem]].
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