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). Furthermore, these representations can be obtained in polynomial time when the planar graph is provided as input. 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. More precisely, we show that 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. Here, \textsc{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. Most of the known NP-hardness reductions for these problems are from variants of 3-SAT. We reduce from \textsc{Planar Hamiltonian Path Completion}, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric graphs.

中文翻译：

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寻找顶点图的几何表示是NP-Hard

平面图可以表示为平面中不同类型的几何对象的交集图，例如圆形（Koebe，1936年），线段（Chalopin \＆Gon {\ c {c}} alves，2009年），\ textsc {L } -shapes（Gon {\ c {c}} alves〜et al。，2018）。此外，当提供平面图作为输入时，可以在多项式时间内获得这些表示。但是，对于一般图形，甚至确定是否存在这种表示形式通常也是NP-hard的。我们考虑顶点图，即可以通过从顶点移除一个顶点而使其变为平面的图。更准确地说，我们表明识别每个满足\ textsc {Pure-2-Dir} $ \ subseteq \ mathcal {G} \ subseteq $ \ textsc {1-String}的图类$ \ mathcal {G} $是NP-hard ，即使输入图是顶点图也是如此。这里，\ textsc {Pure-2-Dir}是轴平行线段的相交图的类（其中水平和垂直线段之间仅允许相交），而\ textsc {1-String}是简单曲线的相交图的类（两条曲线最多共享一个点）。对于这些问题，大多数已知的NP硬度降低都来自3-SAT的变体。我们从\ textsc {Plane Hamiltonian Path Completion}中减少，它使用了更直观的平面性概念。因此，我们的证明要简单得多，并且封装了几类几何图形。对于这些问题，大多数已知的NP硬度降低都来自3-SAT的变体。我们从\ textsc {Plane Hamiltonian Path Completion}中减少，它使用了更直观的平面性概念。因此，我们的证明要简单得多，并且封装了几类几何图形。对于这些问题，大多数已知的NP硬度降低都来自3-SAT的变体。我们从\ textsc {Plane Hamiltonian Path Completion}中减少，它使用了更直观的平面性概念。因此，我们的证明要简单得多，并且封装了几类几何图形。