A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane $\mathbb{R}^2$. Recognizing them is known to be $\exists\mathbb{R}$-complete, i.e., as hard as solving a system of polynomial inequalities. In this note we describe a simple framework to translate $\exists\mathbb{R}$-hardness reductions from the Euclidean plane $\mathbb{R}^2$ to the hyperbolic plane $\mathbb{H}^2$. We apply our framework to prove that the recognition of unit disk graphs in the hyperbolic plane is also $\exists\mathbb{R}$-complete.
翻译：图形 G 是一个单位磁盘图( Euclidean) 单位磁盘图, 如果它是 Euclidean 平面单位磁盘的交叉图 $\ mathbb{R ⁇ 2$, 则是一个单位磁盘图 。 识别它们已知是$\ dexists\ mathb{R}$- 完整的, 也就是像解决多元不平等体系一样困难。 在本说明中, 我们描述一个简单的框架来将 $\ dexistb{ mathb{R} $$- 硬度从 Euclidean 平面 $\ mathb{R} $2$ 转换为超双曲线平面 $\ mathbb{H ⁇ 2$ 。 我们应用我们的框架来证明超单平面单位磁盘图的识别也是$\ dmets\ mathb{ R} $- 完整的 。
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