  Thomassen proved that all planar graphs are $5$-choosable. \v{S}krekovski
strengthened the result by showing that all $K_{5}$-minor-free graphs are
$5$-choosable. Dvo\v{r}\'{a}k and Postle pointed out that all planar graphs are
DP-$5$-colorable. In this note, we first improve these results by showing that
every $K_{5}$-minor-free or $K_{3, 3}$-minor-free graph is DP-$5$-colorable. In
the final section, we further improve these results under the term strictly
$f$-degenerate transversal.
