Finding Geometric Representations of Apex Graphs is NP-Hard

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.