GD 1995: Graph Drawing pp 178-189 | Cite as

The strength of weak proximity (extended abstract)

  • Giuseppe Di Battista
  • Giuseppe Liotta
  • Sue Whitesides
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)

Abstract

This paper initiates the study of weak proximity drawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximity drawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other points representing vertices, then segment (p, q) is allowed, but not forced, to appear in the drawing. This differs from the usual, strong, notion of proximity drawing in which such segments must appear in the drawing.

Most previously studied proximity regions are associated with a parameter β, 0≤β≤∞. For fixed β, weak β-drawability is at least as expressive as strong β-drawability, as a strong β-drawing is also a weak one. We give examples of graph families and β values where the two notions coincide, and a situation in which it is NP-hard to determine weak β-drawability. On the other hand, we give situations where weak proximity significantly increases the expressive power of β-drawability: we show that every graph has, for all sufficiently small β, a weak β-proximity drawing that is computable in linear time, and we show that every tree has, for every β less than 2, a weak β-drawing that is computable in linear time.

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References

  1. 1.
    S. Bhatt and S. Cosmadakis. The Complexity of Minimizing Wire Lengths in VLSI Layouts. Information Processing Letters, 25, 1987, pp. 263–267.CrossRefGoogle Scholar
  2. 2.
    P. Bose, W. Lenhart, and G. Liotta. Characterizing Proximity Trees. Algorithmica, Special Issue on Graph Drawing (to appear).Google Scholar
  3. 3.
    P. Bose, G. Di Battista, W. Lenhart, and G. Liotta. Proximity Constraints and Representable Trees. Graph Drawing, Proc. of the DIMACS International Workshop GD94, New Jersey, USA, October 1994, LNCS 894, R. Tamassia and I.G. Tollis eds., Springer-Verlag, 1995, pp. 340–351.Google Scholar
  4. 4.
    G. Di Battista, W. Lenhart, and G. Liotta. Proximity Drawability: a Survey. Graph Drawing, Proc. of the DIMACS International Workshop GD94, New Jersey, USA, October 1994, LNCS 894, R. Tamassia and I.G. Tollis eds., Springer-Verlag, 1995, pp. 328–339.Google Scholar
  5. 5.
    G. Di Battista, R. Tamassia, and I. G. Tollis. Area Requirement and Symmetry Display of Planar Upward Drawings. Discr. and Comp. Geometry, 7, 1992, pp. 381–401.CrossRefGoogle Scholar
  6. 6.
    P. Eades and S. Whitesides. The Realization Problem for Euclidean Minimum Spanning Trees is NP-hard. Proc. 10th ACM Symposium on Computational Geometry, 1994, pp. 49–56.Google Scholar
  7. 7.
    P. Eades and S. Whitesides. Nearest Neighbor Graph Realizability is NP-hard. Proc. Latin'95, Valparaiso, Chile, 1995, LNCS 911, R. Baeza-Yates, E. Goles, P. Poblete eds., Springer-Verlag, 1995, pp. 245–256.Google Scholar
  8. 8.
    M. Ichino and J. Sklansky. The Relative Neighborhood Graph for Mixed Feature Variables. Pattern Recognition, 18, 1985, pp. 161–167.CrossRefMathSciNetGoogle Scholar
  9. 9.
    G. Kant, G. Liotta, R. Tamassia, and I.G. Tollis. Area Requirement of Visibility Representations of Trees. Proc.5th CCCG, Waterloo, 1993, pp. 192–197.Google Scholar
  10. 10.
    J.M. Keil. Computing a Subgraph of the Minimum Weight Triangulation. Comp. Geom.: Theory and Appl., 4, 1994, pp. 13–26.Google Scholar
  11. 11.
    D. G. Kirkpatrick and J. D. Radke. A Framework for Computational Morphology. Computational Geometry, ed. G. T. Toussaint, Elsevier, Amsterdam, 1985, pp. 217–248.Google Scholar
  12. 12.
    A. Lubiw and N. Sleumer. All Maximal Outerplanar Graphs are Relative Neighborhood Graphs. Proc. 5th CCCG, Waterloo, 1993, pp. 198–203.Google Scholar
  13. 13.
    S. Malitz and A. Papakostas. On the Angular Resolution of Planar Graphs. Proc. STOC, 1993, pp. 431–437.Google Scholar
  14. 14.
    W. Schnyder. Embedding Planar Graphs on the Grid. Proc. SODA, 1990, pp. 41–51.Google Scholar
  15. 15.
    R. Tamassia and I. G. Tollis. A Unified Approach to Visibility Representations of Planar Graphs. Discr. and Comp. Geometry, 1, 1986, pp. 321–341.CrossRefGoogle Scholar
  16. 16.
    M. Formann, T. Hagerup, J. Haralambides, M. Kaufmann, F. T. Leighton, A. Simvonis, E. Welzl, G. Woeginger. Drawing Graphs in the Plane with High Resolution. Proc. FOCS, 1990, pp. 86–95.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Giuseppe Di Battista
    • 1
  • Giuseppe Liotta
    • 2
  • Sue Whitesides
    • 3
  1. 1.Dipartimento di Discipline ScientificheSezione Informatica Terza Universita' di Roma via SegreRoma
  2. 2.Department of Computer ScienceBrown UniversityProvidence
  3. 3.School of Computer ScienceMcGill UniversityMontréalCanada

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