Finding geometric representations of apex graphs is NP-hard

Planar graphs can be represented as the intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., SODA 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 as well. More precisely, we show that for every fixed positive integer g and every graph class G such that [Formula presented], it is NP-hard to decide whether an input graph belongs to the graph class G, even when the inputs are restricted to apex graphs of girth g. Here, [Formula presented] is the class of intersection graphs of axis-parallel line segments (where horizontal segments intersect only vertical segments), and [Formula presented] is the class of intersection graphs of simple curves (where two intersecting curves cross each other exactly once) in the plane. This partially answers an open question raised by Kratochvíl & Pergel (COCOON, 2007). Most known reductions for earlier proofs of NP-hardness for these problems are from variants of 3-SAT (mainly PLANAR 3-CONNECTED 3-SAT). We reduce from the [Formula presented] [Formula presented] [Formula presented] problem, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric intersection graphs.