Advertisement

On the number of directions in visibility representations of graphs (extended abstract)

  • Evangelos Kranakis
  • Danny Krizanc
  • Jorge Urrutia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 894)

Abstract

We consider visibility representations of graphs in which the vertices are represented by a collection O of non-overlapping convex regions on the plane. Two points x and y are visible if the straight-line segment xy is not obstructed by any object. Two objects A, BO are called visible if there exist points xA, yB such that x is visible from y. We consider visibility only for a finite set of directions. In such a representation, the given graph is decomposed into a union of unidirectional visibility graphs, for the chosen set of directions. This raises the problem of studying the number of directions needed to represent a given graph. We study this number of directions as a graph parameter and obtain sharp upper and lower bounds for the represent ability of arbitrary graphs.

1980 Mathematics Subject Classification: 68R10, 68U05 CR Categories: F.2.2

Key words and phrases

Graph Number of directions Polygon Visibility 

References

  1. 1.
    G. Di Battista and P. Eades and R. Tamassia and I. G. Tollis, “Algorithms for Drawing Graphs: An Annotated Bibliography”, Comput. Geom. Theory Appl, to appear. (Preprint available by anonymous ftp from ftp.cs.brown.edu:pub/papers/compgeo/.)Google Scholar
  2. 2.
    P. Bose, H. Everett, S. Fekete, A. Lubiw, H. Meijer, K. Romanik, T. Shermer, and S. Whitesides, “On a Visibility Representation for Graphs in Three Dimensions” (abstract), ALCOM International Workshop on Graph Drawing and Topological Graph Algorithms, Paris, September 26–29, 1993, pp. 53–54. (Also, in McGill Technical Report “Snapshots of Computational and Discrete Geometry”, 1994.)Google Scholar
  3. 3.
    F. Harary, “Graph Theory”, Addison-Wesley Publishing Company, 1969.Google Scholar
  4. 4.
    L. S. Heath and S. Istrail, “The Pagenumber of Genus g Graphs is O(g)”, in STOC87, pages 388–397.Google Scholar
  5. 5.
    J. O'Rourke, “Computational Geometry Column 18”, International Journal of Computational Geometry & Applications, pp. 107–113, Vol. 3, No. 1, 1993.Google Scholar
  6. 6.
    I. Rival and J. Urrutia, “Representing Orders on the Plane by Translating Convex Figures”, Order 4 (1988), 319–329.Google Scholar
  7. 7.
    I. Rival and J. Urrutia, “Representing Orders by Moving Figures in Space”, Discrete Mathematics 109 (1992), 255–263.Google Scholar
  8. 8.
    R. Tamassia and I. G. Tollis, “A Unified Approach to Visibility Representations of Planar Graphs”, Discrete Comput. Geom. 1:321–341, 1986.Google Scholar
  9. 9.
    R. Tamassia and I. G. Tollis, “Plane Representations of Graphs and Visibility Between Parallel Segments”, TR ACT-37, Univ. of Ill., Urbanna Champaign, 1985.Google Scholar
  10. 10.
    H. Warren, “Lower Bounds for Approximation by Nonlinear Manifolds”, Transactions of the AMS 133(1968), 167–178.Google Scholar
  11. 11.
    S. K. Wismath, “Characterizing Bar Line-of-sight Graphs”, Proc. 1st Annu. ACM Sympos. Comput. Geom., pp. 147–152, 1985.Google Scholar
  12. 12.
    M. Yannakakis, “Four Pages are Necessary and Sufficient for Planar Graphs”, in STOC86, pages 104–108.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Evangelos Kranakis
    • 1
  • Danny Krizanc
    • 1
  • Jorge Urrutia
    • 2
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Computer ScienceUniversity of OttawaOttawaCanada

Personalised recommendations