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Kuratowski's theorem
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{{Short description|On forbidden subgraphs in planar graphs}} {{For|the point-set topology theorem|Kuratowski's closure-complement problem}} [[File:GP92-Kuratowski.svg|thumb|240px|A subdivision of ''K''<sub>3,3</sub> in the [[generalized Petersen graph]] ''G''(9,2), showing that the graph is nonplanar.]] In [[graph theory]], '''Kuratowski's theorem''' is a mathematical [[forbidden graph characterization]] of [[planar graph]]s, named after [[Kazimierz Kuratowski]]. It states that a finite graph is planar if and only if it does not contain a [[Glossary of graph theory#Subgraphs|subgraph]] that is a [[subdivision (graph theory)|subdivision]] of <math>K_5</math> (the [[complete graph]] on five [[vertex (graph theory)|vertices]]) or of <math>K_{3,3}</math> (a [[complete bipartite graph]] on six vertices, three of which connect to each of the other three, also known as the [[utility graph]]).
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