  Given a graph $H$ with at least one edge, let $\operatorname{gap}_{H}(n)$
denote the maximum difference between the numbers of edges in two $n$-vertex
edge-maximal graphs with no minor $H$. We show that for exactly four connected
graphs $H$ (with at least two vertices), the class of graphs with no minor $H$
is pure, that is, $\operatorname{gap}_{H}(n) = 0$ for all $n \geq 1$; and for
each connected graph $H$ (with at least two vertices) we have the dichotomy
that either $\operatorname{gap}_{H}(n) = O(1)$ or $\operatorname{gap}_{H}(n) =
\Theta(n)$. Further, if $H$ is 2-connected and does not yield a pure class,
then there is a constant $c>0$ such that $\operatorname{gap}_{H}(n) \sim cn$.
We also give some partial results when $H$ is not connected or when there are
two or more excluded minors.
