  We consider Independent Set and Dominating Set restricted to VPG graphs (or,
equivalently, string graphs). We show that they both remain $\mathsf{NP}$-hard
on $B_0$-VPG graphs admitting a representation such that each grid-edge belongs
to at most one path and each horizontal path has length at most two. On the
other hand, combining the well-known Baker's shifting technique with bounded
mim-width arguments, we provide simple PTASes on VPG graphs admitting a
representation such that each grid-edge belongs to at most $t$ paths and the
length of the horizontal part of each path is at most $c$, for some $c \geq 1$.
