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On Embeddability of Unit Disk Graphs onto Straight Lines

  • Onur ÇağırıcıEmail author
Conference paper
  • 67 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12159)

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.

Notes

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.

References

  1. 1.
    Alber, J., Fiala, J.: Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms 52(2), 134–151 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alomari, A., Aslam, N., Phillips, W., Comeau, F.: Three-dimensional path planning model for mobile anchor-assisted localization in Wireless Sensor Networks. In: 30th IEEE Canadian Conference on Electrical and Computer Engineering, CCECE, pp. 1–5 (2017)Google Scholar
  3. 3.
    Aspnes, J., et al.: A theory of network localization. IEEE Trans. Mob. Comput. 5(12), 1663–1678 (2006)CrossRefGoogle Scholar
  4. 4.
    Balasundaram, B., Butenko, S.: Optimization problems in unit-disk graphs. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, pp. 2832–2844. Springer, Boston (2009).  https://doi.org/10.1007/978-0-387-74759-0CrossRefGoogle Scholar
  5. 5.
    Bhatt, S.N., Cosmadakis, S.S.: The complexity of minimizing wire lengths in VLSI layouts. Inf. Process. Lett. 25(4), 263–267 (1987)CrossRefGoogle Scholar
  6. 6.
    Bonnet, É., Giannopoulos, P., Kim, E.J., Rzążewski, P., Sikora, F.: QPTAS and subexponential algorithm for maximum clique on disk graphs. In: Speckmann, B., Tóth, C.D. (eds.) 34th International Symposium on Computational Geometry, SoCG. LIPIcs, vol. 99, pp. 12:1–12:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)Google Scholar
  7. 7.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Breu, H.: Algorithmic aspects of constrained unit disk graphs. Ph.D. thesis, University of British Columbia (1996)Google Scholar
  9. 9.
    Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Comput. Geom. 9(1–2), 3–24 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Çağırıcı, O.: Exploiting coplanar clusters to enhance 3D localization in wireless sensor networks. Master’s thesis, Izmir University of Economics (2015). http://arxiv.org/abs/1502.07790
  11. 11.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dil, B., Dulman, S., Havinga, P.: Range-based localization in mobile sensor networks. In: Römer, K., Karl, H., Mattern, F. (eds.) EWSN 2006. LNCS, vol. 3868, pp. 164–179. Springer, Heidelberg (2006).  https://doi.org/10.1007/11669463_14CrossRefGoogle Scholar
  13. 13.
    Evans, W., van Garderen, M., Löffler, M., Polishchuk, V.: Recognizing a DOG is hard, but not when it is thin and unit. In: Demaine, E.D., Grandoni, F. (eds.) 8th International Conference on Fun with Algorithms, FUN. LIPIcs, vol. 49, pp. 16:1–16:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)Google Scholar
  14. 14.
    Fishburn, P.C.: Interval Orders and Interval Graphs – A Study on Partially Ordered Sets. Wiley, Hoboken (1985)zbMATHGoogle Scholar
  15. 15.
    Ito, H., Kadoshita, M.: Tractability and intractability of problems on unit disk graphs parameterized by domain area. In: Zhang, X.S., Liu, D.G., Wu, L.Y., Wang, Y. (eds.) Operations Research and Its Applications, 9th International Symposium, ISORA. Lecture Notes in Operations Research, vol. 12, pp. 120–127 (2010)Google Scholar
  16. 16.
    Kang, R.J., Müller, T.: Sphere and dot product representations of graphs. Discrete Comput. Geom. 47(3), 548–568 (2012).  https://doi.org/10.1007/s00454-012-9394-8MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: Unit disk graph approximation. In: Basagni, S., Phillips, C.A. (eds.) Proceedings of the DIALM-POMC Joint Workshop on Foundations of Mobile Computing, pp. 17–23. ACM (2004)Google Scholar
  18. 18.
    McDiarmid, C., Müller, T.: Integer realizations of disk and segment graphs. J. Comb. Theory Ser. B 103(1), 114–143 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Neto, M.F., Goussevskaia, O., dos Santos, V.F.: Connectivity with backbone structures in obstructed wireless networks. Comput. Netw. 127, 266–281 (2017)CrossRefGoogle Scholar
  20. 20.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.) Proceedings of the 10th Annual ACM Symposium on Theory of Computing, STOC, pp. 216–226. ACM (1978)Google Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Masaryk UniversityBrnoCzech Republic

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