Kuratowski's theorem

On forbidden subgraphs in planar graphs

In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K 5 {\displaystyle K_{5}} (the complete graph on five vertices) nor of K 3 , 3 {\displaystyle K_{3,3}} (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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