求顶点图的几何表示是NP困难的
Finding Geometric Representations of Apex Graphs is NP-Hard
摘要：
平面图可以表示为平面中不同类型的几何对象的相交图，例如，圆（Koebe，1936）、线段（Chalopin \& Gon{alves}，2009）、\textsc{L} -形状（Gon{alves} et al，2018）。然而，对于一般的图，甚至决定这样的表示是否存在也常常是$NP$ -困难的。我们考虑顶点图，即，可以通过去掉一个顶点而变成平面的图。我们发现，有点令人惊讶的是，决定是否存在顶点图的几何表示是$NP$ -硬。更准确地说，我们表明，对于每一个正整数$k$，识别每一个图类$\mathcal{G}$满足$\textsc{PURE-2-DIR} \subseteq \mathcal{G} \subseteq \textsc{1-STRING}$是$NP$ -硬，即使当输入的图是顶点图的周长至少$k$。这里，$PURE-2-DIR$是轴平行线段（其中仅允许水平和垂直线段之间的相交）的相交图的类，而\textsc{1-STRING}是平面中简单曲线（其中两条曲线最多共享一个点）的相交图的类。这部分回答了Kratochv l \& Pergel（2007）提出的一个开放性问题。大多数已知的用于这些问题的$NP$硬度降低来自3-SAT的变体。我们减少从\textsc{PLANAR HAMILTONIAN PATH COMPLETION}的问题，它使用更直观的平面性的概念。因此，我们的证明要简单得多，并封装了几类几何图。
英文摘要：
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{ç}alves, 2009), \textsc{L}-shapes (Gon{ç}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{\'ı}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.
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