  Let $H$ be a graph with maximum degree $d$, and let $d'\ge 0$. We show that
for some $c>0$ depending on $H,d'$, and all integers $n\ge 0$, there are at
most $c^n$ unlabelled simple $d$-connected $n$-vertex graphs with maximum
degree at most $d'$ that do not contain $H$ as a subdivision. On the other
hand, the number of unlabelled simple $(d-1)$-connected $n$-vertex graphs with
minimum degree $d$ and maximum degree at most $d+1$ that do not contain
$K_{d+1}$ as a subdivision is superexponential in $n$.
