Discrete & Computational Geometry

, Volume 17, Issue 2, pp 143–162 | Cite as

On recognizing and characterizing visibility graphs of simple polygons

Article

Abstract

In this paper we establish four necessary conditions for recognizing visibility graphs of simple polygons and conjecture that these conditions are sufficient. We present an 0(n2)-time algorithm for testing the first and second necessary conditions and leave it open whether the third and fourth necessary conditions can be tested in polynomial time. We also show that visibility graphs of simple polygons do not possess the characteristics of a few special classes of graphs

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Abello, O. Egecioglu, and K. Kumar, Visibility graphs of staircase polygons and the weak Bruhat order, I: from visibility graphs to maximal chains,Discrete & Computational Geometry, 14(3) (1995), 331–358.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Abello, O. Egecioglu, and K. Kumar, Visibility graphs of staircase polygons and the weak Bruhat order, II: from maximal chains to polygons, preprint.Google Scholar
  3. 3.
    J. Abello and K. Kumar, Visibility graphs and oriented metroids,Proceeding of Graph Drawing, Lecture Notes in Computer Science, Vol.894, Springer-Verlag, Berlin, pp. 147–158, 1995.Google Scholar
  4. 4.
    J. Abello, H. Lin, and S. Pisupati, On visibility graphs of simple polygons,Congressus Numeratium, 90 (1992), 119–128.MathSciNetGoogle Scholar
  5. 5.
    D. Avis and D. Rappaport, Computing the largest empty convex subset of a set of points,Proceedings of the First ACM Symposium on Computational Geometry, pp. 161-167, 1985.Google Scholar
  6. 6.
    M. A. Buckinghan, Circle Graphs, Ph.D. Dissertation, Report No. NSO-21, Courant Institute of Mathematical Sciences, New York, 1980.Google Scholar
  7. 7.
    H. ElGindy, Hierarchical decomposition of polygons with applications, Ph.D. Dissertation, McGill University, Montreal, 1985.Google Scholar
  8. 8.
    H. Everett, Visibility graph recognition, Ph.D. Dissertation, University of Toronto, Toronto, January 1990.Google Scholar
  9. 9.
    H. Everett and D. Corneil, Recognizing visibility graphs of spiral polygons,Journal of Algorithms, 11 (1990), 1–26.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. P. Gabor, W. Hsu, and K. J. Supowit, Recognizing circle graphs in polynomial time,Proceedings of the 26th IEEE Annual Symposium on Foundation of Computer Science, pp. 106-116, 1985.Google Scholar
  11. 11.
    F. Gravil, Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph,SIAM Journal on Computing, 1 (1972), 180–187.CrossRefMathSciNetGoogle Scholar
  12. 12.
    S. K. Ghosh, On recognizing and characterizing visibility graphs of simple polygons, Report JHU/EECS-86/14, The Johns Hopkins University, Baltimore, 1986. Also inProceedings of the Scandinavian Workshop on Algorithm Theory, Lecture Notes in Computer Science, Vol.318, Springer-Verlag, Berlin, pp. 96–104, 1988.Google Scholar
  13. 13.
    S. K. Ghosh, A. Maheshwari, S. P. Pal, S. Saluja, and C. E. Veni Madhavan, Characterizing and recognizing weak visibility polygons,Computational Geometry: Theory and Applications, 3 (1993), 213–233.MATHMathSciNetGoogle Scholar
  14. 14.
    M. C. Golumbic,Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.MATHGoogle Scholar
  15. 15.
    J. Hershberger, An optimal visibility graph algorithm for triangulated simple polygon,Algorithmica, 4 (1989), 141–155.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    T. Lazano-Perez and M. A. Wesley, An algorithm for planning collision free paths among polygonal obstacles,Communications of the ACM, 22 (1979), 560–570.CrossRefGoogle Scholar
  17. 17.
    J. O’Rourke,Art Gallery Theorems and Algorithms, Oxford University Press, Oxford, 1987.MATHGoogle Scholar
  18. 18.
    J. O’Rourke, Computational geometry column 18,SIGACT News, 24 (1993), 20–25.CrossRefGoogle Scholar
  19. 19.
    L. G. Shapiro and R. M. Haralick, Decomposition of two-dimensional shape by graph-theoretic clustering,IEEE Transactions on Pattern Analysis and Machine Intelligence, 1 (1979), 10–19.CrossRefGoogle Scholar
  20. 20.
    T. Shermer, Hiding people in polygons,Computing, 42 (1989), 109–132.CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    G. Srinivasaraghavan and A. Mukhopadhyay, A new necessary condition for the vertex visibility graphs of simple polygons,Discrete & Computational Geometry, 12 (1994), 65–82.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Vertag New York Inc. 1997

Authors and Affiliations

  1. 1.Computer Science GroupTata Institute of Fundamental ResearchBombayIndia

Personalised recommendations