  This paper studies generalizations of the concept of acyclic orientations to
arc-weighted orientations. These lead to four types of variations of strict
degeneracy of graphs. Some of these variations are studied in the literature
under different names and we put them in a same framework for comparison. Then
we concentrate on one of these variations, which is new and is defined as
follows: For a graph $G$ and a mapping $f \in \mathbb{N}^G$, we say $G$ is
  $ST^{(2)}$-$f$-degenerate if there is an arc-weighted orientation $(D, w)$ of
$G$ such that $d_{(D,w)}^+(v) < f(v)$ for each vertex $v$, and every
sub-digraph $D'$ of $D$ contains an arc $e=(u,v)$ with
  $w(e) > d_{(D', w)}^+(v)$.
  We prove that if $G$ is $ST^{(2)}$-$f$-degenerate, then $G$ is $f$-paintable,
as well as $f$-AT.
  Then we use $ST^{(2)}$-degeneracy to study truncated degree choosability of
graphs. A graph $G$ is called $k$-truncated degree-choosable (respectively
$ST^{(2)}$-$k$-truncated degree degenerate) if $G$ is $f$-choosable
(respectively, $ST^{(2)}$-$f$-degenerate), where $f(v)= \min\{k, d_G(v)\}$.
  Richter asked whether every 3-connected non-complete planar graph is
$6$-truncated-degree-choosable. We answer this question in negative by
constructing a 3-connected non-complete planar graph which is not
$7$-truncated-degree-choosable. On the other hand, we prove that every
3-connected non-complete planar graph is
$ST^{(2)}$-$16$-truncated-degree-degenerate, and hence
  $16$-truncated-degree-choosable. We further prove that for an arbitrary
proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum
integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$, then there is a
constant $k$ such that every $s$-connected non-complete graph $G \in {\mathcal
G}$ is $ST^{(2)}$-$k$-truncated-degree-degenerate and hence
$k$-truncated-degree-choosable.
