Unit disk graphs are the intersection graphs of unit radius disks in the Euclidean plane. Deciding whether there exists an embedding of a given unit disk graph, i.e. unit disk graph recognition, is an important geometric problem, and has many application areas. In general, this problem is known to be \(\exists \mathbb {R}\)-complete. In some applications, the objects that correspond to unit disks, have predefined (geometrical) structures to be placed on. Hence, many researchers attacked this problem by restricting the domain of the disk centers. Following the same line, we also describe a polynomial-time reduction which shows that deciding whether a graph can be realized as unit disks onto given straight lines is NP-hard, when the given lines are parallel to either x-axis or y-axis. Adjusting the reduction, we also show that this problem is NP-complete when the given lines are only parallel to x-axis. We obtain those results using the idea of the logic engine introduced by Bhatt and Cosmadakis in 1987.

单位圆盘图是单位半径圆盘在欧氏平面上的交集图。判断给定的单位圆盘图是否存在嵌入，即单位圆盘图识别，是一个重要的几何问题，具有广泛的应用领域。一般来说，已知此问题是\(\exists \mathbb {R}\) -complete 的。在某些应用程序中，对应于单位圆盘的对象具有要放置的预定义（几何）结构。因此，许多研究人员通过限制磁盘中心的域来解决这个问题。沿着同一条线，我们还描述了多项式时间缩减，它表明当给定线平行于x轴或y轴。调整缩减后，我们还表明，当给定线仅平行于x轴时，此问题是 NP 完全问题。我们使用 Bhatt 和 Cosmadakis 在 1987 年引入的逻辑引擎的想法获得了这些结果。