In this paper, we consider the Visibility Graph Recognition and Reconstruction problems in the context of terrains. Here, we are given a graph $G$ with labeled vertices $v_0, v_1, \ldots, v_{n-1}$ such that the labeling corresponds with a Hamiltonian path $H$. $G$ also may contain other edges. We are interested in determining if there is a terrain $T$ with vertices $p_0, p_1, \ldots, p_{n-1}$ such that $G$ is the visibility graph of $T$ and the boundary of $T$ corresponds with $H$. $G$ is said to be persistent if and only if it satisfies the so-called X-property and Bar-property. It is known that every "pseudo-terrain" has a persistent visibility graph and that every persistent graph is the visibility graph for some pseudo-terrain. The connection is not as clear for (geometric) terrains. It is known that the visibility graph of any terrain $T$ is persistent, but it has been unclear whether every persistent graph $G$ has a terrain $T$ such that $G$ is the visibility graph of $T$. There actually have been several papers that claim this to be the case (although no formal proof has ever been published), and recent works made steps towards building a terrain reconstruction algorithm for any persistent graph. In this paper, we show that there exists a persistent graph $G$ that is not the visibility graph for any terrain $T$. This means persistence is not enough by itself to characterize the visibility graphs of terrains, and implies that pseudo-terrains are not stretchable.

中文翻译：

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地形可见性图：持久性是不够的

在本文中，我们考虑了地形背景下的可见性图识别和重建问题。在这里，我们给出了一个带有标记顶点 $v_0、v_1、\ldots、v_{n-1}$ 的图 $G$，这样标记对应于哈密顿路径 $H$。$G$ 也可能包含其他边。我们感兴趣的是确定是否有一个地形 $T$ 的顶点 $p_0, p_1, \ldots, p_{n-1}$ 使得 $G$ 是 $T$ 的可见性图和 $T$ 的边界对应于$H$。当且仅当它满足所谓的 X-property 和 Bar-property 时，才称 $G$ 是持久的。众所周知，每个“伪地形”都有一个持久可见性图，并且每个持久性图都是某个伪地形的可见性图。对于（几何）地形，这种联系不是很清楚。已知任何地形$T$的能见度图都是持久的，但不清楚每个持久图$G$是否都有地形$T$使得$G$是$T$的能见度图。实际上有几篇论文声称情况确实如此（尽管从未发表过正式的证据），并且最近的工作朝着为任何持久图构建地形重建算法迈出了一步。在本文中，我们表明存在一个持久图 $G$，它不是任何地形 $T$ 的可见性图。这意味着持久性本身不足以表征地形的可见性图，并暗示伪地形不可拉伸。实际上有几篇论文声称情况确实如此（尽管从未发表过正式的证据），并且最近的工作朝着为任何持久图构建地形重建算法迈出了一步。在本文中，我们表明存在一个持久图 $G$，它不是任何地形 $T$ 的可见性图。这意味着持久性本身不足以表征地形的可见性图，并暗示伪地形不可拉伸。实际上有几篇论文声称情况确实如此（尽管从未发表过正式的证据），并且最近的工作朝着为任何持久图构建地形重建算法迈出了一步。在本文中，我们表明存在一个持久图 $G$，它不是任何地形 $T$ 的可见性图。这意味着持久性本身不足以表征地形的可见性图，并暗示伪地形不可拉伸。