  It is currently an unsolved problem to determine whether a $\triangle$-free
planar graph $G$ contains an independent set $A$ such that $G[V_G\setminus A]$
is $2$-choosable. However, in this paper, we take a slightly different approach
by relaxing the planarity condition. We prove the $\mathbb{NP}$-completeness of
the above decision problem when the graph is $\triangle$-free, $4$-colorable,
and of diameter $3$. Building upon this notion, we examine the computational
complexity of two optimization problems: minimum near $3$-choosability and
minimum $2$-choosable deletion. In the former problem, the goal is to find an
independent set $A$ of minimum size in a given graph $G$, such that the induced
subgraph $G[V_G \setminus A]$ is $2$-choosable. We establish that this problem
is $\mathbb{NP}$-hard to approximate within a factor of $|V_G|^{1-\epsilon}$
for any $\epsilon > 0$, even for planar bipartite graphs. On the other hand,
the problem of minimum $2$-choosable deletion involves determining a vertex set
$A \subseteq V_G$ of minimum cardinality such that the induced subgraph $G[V_G
\setminus A]$ is $2$-choosable. We prove that this problem is
$\mathbb{NP}$-complete, but can be approximated within a factor of $O(\log
|V_G|)$.
