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On Colouring Point Visibility Graphs

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Algorithms and Discrete Applied Mathematics (CALDAM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10156))

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Abstract

In this paper we show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4.

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References

  1. Cardinal, J., Hoffmann, U.: Recognition and complexity of point visibility graphs. In: Symposium of Computational Geometry, pp. 171–185 (2015)

    Google Scholar 

  2. Chazelle, B., Guibas, L.J., Lee, D.T.: The power of geometric duality. BIT 25, 76–90 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cibulka, J., Kyncl, J., Valtr, P.: On planar point sets with the pentagon property. In: Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry, pp. 81–90 (2013)

    Google Scholar 

  4. De Berg, M., Cheong, O., Kreveld, M., Overmars, M.: Computational Geometry, Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  5. Diwan, A.A., Roy, B.: Partitions of planar point sets into polygons. In: Proceedings of the 28th Canadian Conference on Computational Geometry, pp. 147–154 (2016)

    Google Scholar 

  6. Edelsbrunner, H., O’Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15, 341–363 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H Freeman and Company, New York (1979)

    MATH  Google Scholar 

  8. Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)

    Book  MATH  Google Scholar 

  9. Ghosh, S.K., Goswami, P.P.: Unsolved problems in visibility graphs of points, segments and polygons. ACM Comput. Surv. 46(2), 22:1–22:29 (2013)

    Article  MATH  Google Scholar 

  10. Ghosh, S.K., Roy, B.: Some results on point visibility graphs. Theor. Comput. Sci. 575, 17–32 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kára, J., Pór, A., Wood, D.R.: On the chromatic number of the visibility graph of a set of points in the plane. Discret. Comput. Geom. 34(3), 497–506 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lozano-Perez, T., Wesley, M.A.: An algorithm for planning collision-free paths among polyhedral obstacles. Commun. ACM 22, 560–570 (1979)

    Article  Google Scholar 

  13. Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)

    MathSciNet  MATH  Google Scholar 

  14. Payne, M.S., Pór, A., Valtr, P., Wood, D.R.: On the connectivity of visibility graphs. Discret. Comput. Geom. 48(3), 669–681 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pfender, F.: Visibility graphs of point sets in the plane. Discret. Comput. Geom. 39(1), 455–459 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Roy, B.: Point visibility graph recognition is NP-hard. Int. J. Comput. Geom. Appl. 26(1), 1–32 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shapiro, L.G., Haralick, R.M.: Decomposition of two-dimensional shape by graph-theoretic clustering. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1, 10–19 (1979)

    Google Scholar 

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Acknowledgements

The authors are grateful to the referees for their valuable comments and specially pointing out reference [15].

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

    Ajit Arvind Diwan & Bodhayan Roy

Authors
  1. Ajit Arvind Diwan
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  2. Bodhayan Roy
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Corresponding author

Correspondence to Bodhayan Roy .

Editor information

Editors and Affiliations

  1. University of Lethbridge, Lethbridge, Alberta, Canada

    Daya Gaur

  2. Indian Institute of Technology Madras, Chennai, India

    N.S. Narayanaswamy

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Diwan, A.A., Roy, B. (2017). On Colouring Point Visibility Graphs. In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_14

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  • DOI: https://doi.org/10.1007/978-3-319-53007-9_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53006-2

  • Online ISBN: 978-3-319-53007-9

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