Abstract
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 the 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 the x-axis (and one another). We obtain these results using the idea of the logic engine introduced by Bhatt and Cosmadakis in 1987.
This work is supported by the Czech Science Foundation, project no. 20-04567S.
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Notes
- 1.
This problem is equivalent to the 2-coloring of 3-uniform hypergraphs. We choose to give the reduction from Monotone NAE3SAT as it is more intuitive to construct for our problem.
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Acknowledgments
The author wants to thank Petr Hliněný for his insight on the hardness proof. In addition, he thanks Deniz Ağaoğlu and Michał Dębski for their extensive comments and generous help during the preparation of this manuscript.
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Çağırıcı, O. (2020). On Embeddability of Unit Disk Graphs onto Straight Lines. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_13
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