Finding Geometric Representations of Apex Graphs is NP-Hard

Abstract

AbstractPlanar graphs can be represented as 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 \(\mathsf{NP}\)-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 \(\mathsf{NP}\)-hard.More precisely, we show that for every positive integer g, recognizing every graph class \(\mathscr {G}\) such that \({\textsc {Pure}}\text {-}{\textsc {2}}\text {-}{\textsc {Dir}}\subseteq \mathscr {G}\subseteq {\textsc {1}}\text {-}{\textsc {String}}\) is \(\mathsf{NP}\)-hard, even if the inputs are apex graphs of girth at least g. Here, \({\textsc {Pure}}\text {-}{\textsc {2}}\text {-}{\textsc {Dir}}\) is the class of intersection graphs of axis-parallel line segments (where intersections are allowed only between horizontal and vertical segments), and \({\textsc {1}}\text {-}{\textsc {String}}\) is the class of intersection graphs of simple curves (where two curves cross at most once) in the plane. This partially answers an open question raised by Kratochvíl & Pergel (COCOON, 2007).Most known \(\textsf {NP}\)-hardness reductions for these problems are from variants of 3-SAT. We reduce from the 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.KeywordsHamiltonian pathplanar graphapex graph\(\mathsf{NP}\)-hardrecognition problemgeometric intersection graphVLSI design1-STRINGPURE-2-DIR