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On Arrangements of Orthogonal Circles

  • Steven Chaplick
  • Henry Förster
  • Myroslav KryvenEmail author
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

Abstract

In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have only a linear number of faces. This implies that orthogonal circle intersection graphs have only a linear number of edges. When we restrict ourselves to orthogonal unit circles, the resulting class of intersection graphs is a subclass of penny graphs (that is, contact graphs of unit circles). We show that, similarly to penny graphs, it is NP-hard to recognize orthogonal unit circle intersection graphs.

Notes

Acknowledgments

We thank Alon Efrat for useful discussions and an anonymous reviewer for pointing us to the Gauss-Bonnet formula.

References

  1. 1.
    Agarwal, P.K., Aronov, B., Sharir, M.: On the complexity of many faces in arrangements of pseudo-segments and circles. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry: The Goodman-Pollack Festschrift, pp. 1–24. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-642-55566-4_1CrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Last, H., Pinchasi, R., Sharir, M.: On the complexity of arrangements of circles in the plane. Discret. Comput. Geom. 26(4), 465–492 (2001).  https://doi.org/10.1007/s00454-001-0043-xMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Comput. Geom. Theory Appl. 9(1–2), 3–24 (1998).  https://doi.org/10.1016/S0925-7721(97)00014-XMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cerioli, M.R., Faria, L., Ferreira, T.O., Protti, F.: A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation. RAIRO Theor. Inf. Appl. 45(3), 331–346 (2011).  https://doi.org/10.1051/ita/2011106MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chaplick, S., Förster, H., Kryven, M., Wolff, A.: On arrangements of orthogonal circles. ArXiv report (2019). https://arxiv.org/abs/1907.08121
  6. 6.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discret. Math. 86(1–3), 165–177 (1990).  https://doi.org/10.1016/0012-365X(90)90358-OMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Upper Saddle River (1999)zbMATHGoogle Scholar
  8. 8.
    Dumitrescu, A., Pach, J.: Minimum clique partition in unit disk graphs. Graphs Combin. 27(3), 399–411 (2011).  https://doi.org/10.1007/s00373-011-1026-1MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eades, P., Whitesides, S.: The logic engine and the realization problem for nearest neighbor graphs. Theoret. Comput. Sci. 169(1), 23–37 (1996).  https://doi.org/10.1016/S0304-3975(97)84223-5MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eppstein, D.: Circles crossing at equal angles (2018). https://11011110.github.io/blog/2018/12/22/circles-crossing-equal.html. Accessed 11 May 2019
  11. 11.
    Eppstein, D.: Triangle-free penny graphs: degeneracy, choosability, and edge count. In: Frati, F., Ma, K.L. (eds.) GD 2017. LNCS, vol. 10692. Springer, Heidelberg (2018).  https://doi.org/10.1007/978-3-319-73915-1_39. https://arxiv.org/abs/1708.05152CrossRefGoogle Scholar
  12. 12.
    Felsner, S.: Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-322-80303-0CrossRefzbMATHGoogle Scholar
  13. 13.
    Felsner, S., Scheucher, M.: Arrangements of pseudocircles: triangles and drawings. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 127–139. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-73915-1_11CrossRefzbMATHGoogle Scholar
  14. 14.
    Füredi, Z., Palásti, I.: Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. 92(4), 561–566 (1984).  https://doi.org/10.1090/S0002-9939-1984-0760946-2MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grünbaum, B.: Arrangements and spreads. In: CBMS Regional Conference Series in Mathmatics, vol. 10. AMS, Providence (1972)Google Scholar
  16. 16.
    Gyárfás, A., Hubenko, A., Solymosi, J.: Large cliques in \({C}_4\)-free graphs. Combinatorica 22(2), 269–274 (2002).  https://doi.org/10.1007/s004930200012MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hliněný, P., Kratochvíl, J.: Representing graphs by disks and balls (a survey of recognition complexity results). Discret. Math. 229(1–3), 101–124 (2001).  https://doi.org/10.1016/S0012-365X(00)00204-1MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kang, R.J., Müller, T.: Sphere and dot product representations of graphs. Discret. Comput. Geom. 47(3), 548–568 (2012).  https://doi.org/10.1007/s00454-012-9394-8MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kang, R.J., Müller, T.: Arrangements of pseudocircles and circles. Discret. Comput. Geom. 51(4), 896–925 (2014).  https://doi.org/10.1007/s00454-014-9583-8MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ogilvy, C.S.: Excursions in Geometry. Oxford University Press, New York (1969)zbMATHGoogle Scholar
  21. 21.
    Pach, J., Agarwal, K.P.: Combinatorial Geometry. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Hoboken (1995)CrossRefGoogle Scholar
  22. 22.
    Pinchasi, R.: Gallai-Sylvester theorem for pairwise intersecting unit circles. Discret. Comput. Geom. 28(4), 607–624 (2002).  https://doi.org/10.1007/s00454-002-2892-3MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Steiner, J.: Einige Gesetze über die Theilung der Ebene und des Raumes. Journal für die reine und angewandte Mathematik 1, 349–364 (1826).  https://doi.org/10.1515/crll.1826.1.349MathSciNetCrossRefGoogle Scholar
  24. 24.
    Weisstein, E.W.: Gauss-Bonnet formula (2019). http://mathworld.wolfram.com/Gauss-BonnetFormula.html. Accessed 27 July 2019

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Universität WürzburgWürzburgGermany
  2. 2.Universität TübingenTübingenGermany

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