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Kuratowski's theorem

A graph is planar exactly when it doesn't secretly contain a stretched-out copy of the two smallest non-planar graphs, K5 and K3,3.

Kuratowski's theorem: $G$ is planar if and only if $G$ contains no subgraph that is a subdivision of $K_5$ or $K_{3,3}$ (theorem: kuratowskis-theorem).