Finding Geometric Representations of Apex Graphs is NP-Hard

Planar graphs can be represented as the intersection of different types geometric objects in plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., 2018). For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex i.e., that made planar by removing one vertex from them. show, somewhat surprisingly, for NP-hard well. More precisely, we show every fixed positive integer g and graph class G , it to decide an input belongs G, when inputs are restricted girth g. Here, axis-parallel (where horizontal intersect only vertical segments), simple curves two intersecting cross each other exactly once) plane. This partially answers open question raised Kratochvíl Pergel (COCOON, 2007). Most known reductions earlier proofs NP-hardness these problems variants 3-SAT (mainly 3-Connected 3-SAT). reduce problem, which uses more intuitive notion planarity. As a result, our proof much simpler encapsulates several classes graphs.