  Grid graphs, and, more generally, $k\times r$ grid graphs, form one of the
most basic classes of geometric graphs. Over the past few decades, a large body
of works studied the (in)tractability of various computational problems on grid
graphs, which often yield substantially faster algorithms than general graphs.
Unfortunately, the recognition of a grid graph is particularly hard -- it was
shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this
paper, we provide several positive results in this regard in the framework of
parameterized complexity (additionally, we present new and complementary
hardness results). Specifically, our contribution is threefold. First, we show
that the problem is fixed-parameter tractable (FPT) parameterized by $k+\mathsf
{mcc}$ where $\mathsf{mcc}$ is the maximum size of a connected component of
$G$. This also implies that the problem is FPT parameterized by $\mathtt{td}+k$
where $\mathtt{td}$ is the treedepth of $G$ (to be compared with the hardness
for pathwidth 2 where $k=3$). Further, we derive as a corollary that strip
packing is FPT with respect to the height of the strip plus the maximum of the
dimensions of the packed rectangles, which was previously only known to be in
XP. Second, we present a new parameterization, denoted $a_G$, relating graph
distance to geometric distance, which may be of independent interest. We show
that the problem is para-NP-hard parameterized by $a_G$, but FPT parameterized
by $a_G$ on trees, as well as FPT parameterized by $k+a_G$. Third, we show that
the recognition of $k\times r$ grid graphs is NP-hard on graphs of pathwidth 2
where $k=3$. Moreover, when $k$ and $r$ are unrestricted, we show that the
problem is NP-hard on trees of pathwidth 2, but trivially solvable in
polynomial time on graphs of pathwidth 1.

Data Structures and Algorithms