Wagner’s theorem

Theorem 1 (Wagner)

A graph is planar if and only if it contains neither $K_{\mathrm{5}}$ nor $K_{\mathrm{3}\mathrm{,}\mathrm{3}}$ as a minor, where $K_{\mathrm{5}}$ is the complete graph of order 5 and $K_{\mathrm{3}\mathrm{,}\mathrm{3}}$ is the complete bipartite graph of order 6.

Wagner’s theorem is to Kuratowski’s theorem.