  We prove that the complement of any non-separating planar graph of order
$2n-3$ contains a $K_n$ minor, and argue that the order $2n-3$ is lowest
possible with this property. To illustrate the necessity of the non-separating
hypothesis, we give an example of a planar graph of order 11 whose complement
does not contain a $K_7$ minor. We argue that the complements of planar graphs
of order 11 are intrinsically knotted. We compute the Hadwiger numbers of
complements of wheel graphs.
