  Motivated by Hadwiger's conjecture and related problems for list-coloring, we
study graphs $H$ for which every graph with minimum degree at least $|V(H)|-1$
contains $H$ as a minor. We prove that a large class of apex-outerplanar graphs
satisfies this property. Our result gives the first examples of such graphs
whose vertex cover numbers are significantly larger than half of the number of
its vertices, which breaks a barrier for attacking related coloring problems
via extremal functions, and recovers all known such graphs that have
arbitrarily large maximum degree. Our proof can be adapted to directed graphs
to show that if $\vec H$ is the digraph obtained from a directed cycle or an
in-arborescence by adding an apex source, then every digraph with minimum
out-degree $|V(\vec H)|-1$ contains $\vec H$ as a subdivision or a butterfly
minor respectively. These results provide the optimal upper bound for the
chromatic number and dichromatic number of graphs and digraphs that do not
contain the aforementioned graphs or digraphs as a minor, butterfly minor and a
subdivision, respectively. Special cases of our results solve an open problem
of Aboulker, Cohen, Havet, Lochet, Moura and Thomass\'{e} and strengthen
results of Gishboliner, Steiner and Szab\'{o}.
