On the purity of minor-closed classes of graphs

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.