  In this paper, we determine the computational complexity of recognizing two
graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection}
graphs. An L-shape is made by joining the bottom end-point of a vertical
($\vert$) segment to the left end-point of a horizontal ($-$) segment. The top
end-point of the vertical segment is known as the {\em anchor} of the L-shape.
Grounded L-graphs are the intersection graphs of L-shapes such that all the
L-shapes' anchors lie on the same horizontal line. We show that recognizing
grounded L-graphs is NP-complete. This answers an open question asked by
Jel{\'\i}nek \& T{\"o}pfer (Electron. J. Comb., 2019).
  Grid intersection graphs are the intersection graphs of axis-parallel line
segments in which two vertical (similarly, two horizontal) segments cannot
intersect. We say that a (not necessarily axis-parallel) straight line $\ell$
stabs a segment $s$, if $s$ intersects $\ell$. A graph $G$ is a stabbable grid
intersection graph ($StabGIG$) if there is a grid intersection representation
of $G$ in which the same line stabs all its segments. We show that recognizing
$StabGIG$ graphs is $NP$-complete, even on a restricted class of graphs. This
answers an open question asked by Chaplick \etal (\textsc{O}rder, 2018).
