  Bernshteyn and Lee defined a new notion, weak degeneracy, which is slightly
weaker than the ordinary degeneracy. It is proved that strictly $f$-degenerate
transversal is a common generalization of list coloring, $L$-forested-coloring
and DP-coloring. In this paper, we consider three classes of graphs, including
planar graphs without any configuration in Fig. 2, toroidal graphs without any
configuration in Fig. 5, and planar graphs without intersecting $5$-cycles. We
give structural results for each class of graphs, and prove each structure is
reducible for weakly $3$-degenerate and the existence of strictly
$f$-degenerate transversals. As consequences, these three classes of graphs are
weakly $3$-degenerate, and have a strictly $f$-degenerate transversal. Then
these three classes of graph have DP-paint number at most four, and have list
vertex arboricity at most two. This greatly improve all the results in [2-4,
11-13, 16-18, 22, 25, 32, 34]. Furthermore, the first and the third classes of
graphs have Alon-Tarsi number at most four.
