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A graph $H$ is called a minor of a graph $G$ if $H$ can be obtained from $G$ by contracting edges, deleting edges, and deleting isolated vertices.

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Does every $4$-connected nonplanar graph contain a $K_5$-minor?

By Kuratowski's theorem, every nonplanar graph contains a (topological) minor of $K_5$ or $K_{3,3}$. But I observed that every time I construct a $4$-connected nonplanar graph, it always contains not …
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