Rectangle-visibility representations of bipartite graphs

  • Alice M. Dean
  • Joan P. Hutchinson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 894)

Abstract

The paper considers representations of bipartite graphs as rectanglevisibility graphs, i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. It is shown that, for p≤q, Kp, q has a representation with no rectangles having collinear sides if and only if p≤3 or p=3 and q≤4. More generally, it is shown that Kp, q is a rectangle-visibility graph if and only if p≤4. Finally, it is shown that every bipartite rectangle-visibility graph on n≥4 vertices has at most 4n−12 edges.

References

  1. 1.
    T. Andreae, Some results on visibility graphs, Disc. Appl. Math. 40 (1992), 5–18.Google Scholar
  2. 2.
    L. W. Beineke, F. Harary, and J. W. Moon, On the thickness of the complete bipartite graph, Proc. Cambridge Philo. Soc. 60 (1964), 1–5.Google Scholar
  3. 3.
    P. Bose, A. Josefczyk, J. Miller, and J. O'Rourke, K 42 is a box visibility graph, Tech. Report #034, Smith College (1994).Google Scholar
  4. 4.
    M. R. Garey, D. S. Johnson, and H. C. So, An application of graph coloring to printed circuit testing, IEEE Trans. Circuits and Systems CAS-23 (1976), 591–599.Google Scholar
  5. 5.
    J. Hopcroft and R. Tarjan, Efficient planarity testing, J. Assoc. Comput. Mach. 21 (1974), 549–568.Google Scholar
  6. 6.
    J. P. Hutchinson, Coloring ordinary maps, maps of empires, and maps of the Moon, Mathematics Magazine 66 (1993), 211–226.Google Scholar
  7. 7.
    J. P. Hutchinson, T. Shermer, and A. Vince, Representations of thickness two graphs, preprint.Google Scholar
  8. 8.
    D. G. Kirkpatrick and S. K. Wismath, Weighted visibility graphs of bars and related flow problems, Lecture Notes in Computer Science (Proc. 1st Workshop Algorithms Data Struct.), vol. 382, Springer-Verlag, 1989, pp. 325–334.Google Scholar
  9. 9.
    F. Luccio, S. Mazzone, and C. K. Wong, A note on visibility graphs, Disc. Math. 64 (1987), 209–219.Google Scholar
  10. 10.
    A. Mansfield, Determining the thickness of graphs is NP-hard, Math. Proc. Camb. Phil. Soc. 93 (1983), 9–23.Google Scholar
  11. 11.
    J. O'Rourke, Art Gallery Theorems and Algorithms, Oxford University Press, N.Y., 1987.Google Scholar
  12. 12.
    R. Tamassia and I.G. Tollis, A unified approach to visibility representations of planar graphs, Disc. and Comp. Geom. 1 (1986), 321–341.Google Scholar
  13. 13.
    S. K. Wismath, Characterizing bar line-of-sight graphs, Proc. 1st Symp. Comp. Geom., ACM (1985), 147–152.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Alice M. Dean
    • 1
  • Joan P. Hutchinson
    • 2
  1. 1.Department of Mathematics and Computer ScienceSkidmore CollegeSaratoga Springs
  2. 2.Department of Mathematics and Computer ScienceMacalester CollegeSt. Paul

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