  Motivated by Hadwiger's conjecture, Seymour asked which graphs $H$ have the
property that every non-null graph $G$ with no $H$ minor has a vertex of degree
at most $|V(H)|-2$. We show that for every monotone graph family $\mathcal{F}$
with strongly sublinear separators, all sufficiently large bipartite graphs $H
\in \mathcal{F}$ with bounded maximum degree have this property. None of the
conditions that $H$ belongs to $\mathcal{F}$, that $H$ is bipartite and that
$H$ has bounded maximum degree can be omitted.
