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  "body": "\n## Statement\n\nA subdivision of a [[graph|0JD5C82QMB]] $H$ is a graph that $H$ gives when some edges are replaced by paths with new inner vertices.\nLet $G$ be a finite graph.\nThen $G$ is [[planar|029JQEK15B]] if and only if no [[subgraph|0P11X5HKAX]] of $G$ is a subdivision of $K_5$ or of $K_{3,3}$.\n\n## Remarks\n\nOne direction is easy: a subdivision of a nonplanar graph is nonplanar, and [[$K_5$ and $K_{3,3}$ are not planar|030A3FHN14]].\nWagner's theorem is a related result: a finite graph is planar if and only if it has neither $K_5$ nor $K_{3,3}$ as a minor.\n\n## History\n\nKazimierz Kuratowski published the theorem in 1930.\nOrrin Frink and Paul Smith proved it independently in the same year, but they did not publish their proof.\nLev Pontryagin reportedly proved it around 1927, and in the Soviet Union it was also known under his name.\n\n## Explanations\n\n### L1 human\n\nThere are exactly two kinds of obstacle to drawing a graph without crossings.\nA graph is planar unless it contains a copy of $K_5$ or $K_{3,3}$ in which some edges are stretched into paths.\n",
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