Kuratowski's theorem

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 ${\displaystyle K_{5}}$ (the complete graph on five vertices) or of ${\displaystyle K_{3,3}}$ (a complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph).