  Planar graphs can be represented as intersection graphs of different types of
geometric objects in the plane, e.g., circles (Koebe, 1936), line segments
(Chalopin \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al,
2018). For general graphs, however, even deciding whether such representations
exist is often $NP$-hard. We consider apex graphs, i.e., graphs that can be
made planar by removing one vertex from them. We show, somewhat surprisingly,
that deciding whether geometric representations exist for apex graphs is
$NP$-hard.
  More precisely, we show that for every positive integer $k$, recognizing
every graph class $\mathcal{G}$ which satisfies $\textsc{PURE-2-DIR} \subseteq
\mathcal{G} \subseteq \textsc{1-STRING}$ is $NP$-hard, even when the input
graphs are apex graphs of girth at least $k$. Here, $PURE-2-DIR$ is the class
of intersection graphs of axis-parallel line segments (where intersections are
allowed only between horizontal and vertical segments) and \textsc{1-STRING} is
the class of intersection graphs of simple curves (where two curves share at
most one point) in the plane. This partially answers an open question raised by
Kratochv{\'\i}l \& Pergel (2007).
  Most known $NP$-hardness reductions for these problems are from variants of
3-SAT. We reduce from the \textsc{PLANAR HAMILTONIAN PATH COMPLETION} problem,
which uses the more intuitive notion of planarity. As a result, our proof is
much simpler and encapsulates several classes of geometric graphs.
