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. One example to such applications is wireless sensor networks, where each disk corresponds to a wireless sensor node, and a pair of intersecting disks corresponds to a pair of sensors being able to communicate with one another. It is usually assumed that the nodes have identical sensing ranges, and thus a unit disk graph model is used to model problems concerning wireless sensor networks. We consider the unit disk graph realization problem on a restricted domain, by assuming a scenario where the wireless sensor nodes are deployed on the corridors of a building. Based on this scenario, we impose a geometric constraint such that the unit disks must be centered onto given straight lines. In this paper, we first 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. Using the reduction we described, we also show that this problem is NP-complete when the given lines are only parallel to $x$-axis (and one another). We obtain those results using the idea of the logic engine introduced by Bhatt and Cosmadakis in 1987.

中文翻译：

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关于单位圆盘图在直线上的嵌入性

单位圆盘图是单位半径圆盘在欧几里得平面上的交集图。判断一个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，有很多应用领域。一般来说，这个问题已知为 $\exists\mathbb{R}$-complete。在某些应用程序中，对应于单元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。此类应用的一个示例是无线传感器网络，其中每个磁盘对应一个无线传感器节点，一对相交的磁盘对应一对能够相互通信的传感器。通常假设节点具有相同的感应范围，因此，单位圆盘图模型用于对有关无线传感器网络的问题进行建模。我们通过假设无线传感器节点部署在建筑物的走廊上的场景来考虑受限域上的单元盘图实现问题。基于这种情况，我们施加了几何约束，使得单位圆盘必须在给定的直线上居中。在本文中，我们首先描述了多项式时间减少，这表明当给定的线平行于 $x$ 轴或 $y 时，决定一个图是否可以实现为给定直线上的单位圆盘是 NP-hard $-轴。使用我们描述的减少，我们还表明，当给定的线仅平行于 $x$ 轴（并且彼此平行）时，这个问题是 NP 完全的。