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

Computing Planar Intertwines

The proof of Wagner's Conjecture by Robertson and Seymour gives a finite description of any family of graphs which is closed under the minor ordering. This description is a finite set of graphs called the obstructions of the family. Since the intersection and the union of two minor closed graph families is again a minor closed graph family, an interesting question is that of computing the obstructions of the new family given the obstructions for the original two families. It is easy to compute the obstructions of the intersection but, until very recently, it was an open problem to compute the obstructions of the union. We show that if the original families are planar then the obstructions of the union are no larger than n O(n 2 ) where n is the size of the largest obstruction of the original family. 1 Introduction Robertson and Seymour's proof of Wagner's conjecture [RSb] raises somes interesting computational questions. An immediate corollary of the theorem is that for every fami..