en:Category:Mathematics images: Date: 15 September 2006: Source: Originally uploaded as png to en.wikipedia description page is/was here. 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). A collection of unit circle and the corresponding unit disk graph. 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. (The closed unit disk is similarly defined as. Based on this scenario, we impose a geometric constraint such that the unit disks must be centered onto given straight lines. The (open) unit disk can also be considered to be the region in the complex plane defined by, where denotes the complex modulus. 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. 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. 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. Hence, many researchers attacked this problem by restricting the domain of the disk centers. In some applications, the objects that correspond to unit disks, have predefined (geometrical) structures to be placed on. In general, this problem is known to be $\exists\mathbb$-complete. unit disk graph recognition, is an important geometric problem, and has many application areas. Deciding whether there exists an embedding of a given unit disk graph, i.e. Unit disk graphs are the intersection graphs of unit radius disks in the Euclidean plane.
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