# 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 & Gonçalves, 2009), L-shapes (Gonçalves 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 𝒢 which satisfies PURE-2-DIR⊆𝒢⊆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 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 Kratochvíl & Pergel (2007). Most known 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.

## Authors

• 7 publications
• 5 publications
• ### Planar graphs as L-intersection or L-contact graphs

The L-intersection graphs are the graphs that have a representation as i...
07/27/2017 ∙ by Daniel Gonçalves, et al. ∙ 0

• ### Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability

We study the problem of finding a maximum cardinality minimal separator ...
09/25/2020 ∙ by Tesshu Hanaka, et al. ∙ 0

• ### Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors

Suppose we have an arrangement A of n geometric objects x_1, ..., x_n ⊆R...
12/20/2017 ∙ by Katharina Klost, et al. ∙ 0

• ### Rigid Foldability is NP-Hard

In this paper, we show that deciding rigid foldability of a given crease...
12/04/2018 ∙ by Hugo Akitaya, et al. ∙ 0

• ### Approximating Dominating Set on Intersection Graphs of L-frames

We consider the Dominating Set (DS) problem on the intersection graphs o...
03/16/2018 ∙ by Sayan Bandyapadhyay, et al. ∙ 0

• ### Axes-parallel unit disk graph recognition is NP-hard

Unit disk graphs are the intersection graphs of unit diameter disks in t...
11/24/2018 ∙ by Onur Çağırıcı, et al. ∙ 0