On representations of some thickness-two graphs

Extended abstract
  • Joan P. Hutchinson
  • Thomas Shermer
  • Andrew Vince
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)

Abstract

This paper considers representations of graphs as rectanglevisibility graphs and as doubly linear graphs. These are, respectively, graphs whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical visibility, and graphs that can be drawn as the union of two straight-edged planar graphs. We prove that these graphs have, with n vertices, at most 6n−20 (resp., 6n−18) edges, and we provide examples of these graphs with 6n−20 edges for each n≥8.

References

  1. 1.
    J. Battle, F. Harary, and Y. Kodama, Every planar graph with nine points has a nonplanar complement, Bull. Amer. Math. Soc. 68 (1962) 569–571.Google Scholar
  2. 2.
    A. Dean and J. P. Hutchinson, Rectangle-visibility representations of bipartite graphs, Extended Abstract, Lecture Notes in Computer Science (Proc. DIMACS Workshop Graph Drawing, 1994), R. Tamassia and I. G. Tollis, eds., vol. 894, Springer-Verlag, 1995, 159–166.Google Scholar
  3. 3.
    -, Rectangle-visibility representations of bipartite graphs (submitted).Google Scholar
  4. 4.
    M. Gardner, Mathematical Games, Scientific American 242 (Feb. 1980) 14–19.Google Scholar
  5. 5.
    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.CrossRefGoogle Scholar
  6. 6.
    J. P. Hutchinson, Coloring ordinary maps, maps of empires, and maps of the Moon, Math. Mag. 66 (1993) 211–226.Google Scholar
  7. 7.
    J. P. Hutchinson, T. Shermer, and A. Vince, On representations of some thickness-two graphs (submitted).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, 325–334.Google Scholar
  9. 9.
    A. Mansfield, Determining the thickness of graphs is NP-hard, Math. Proc. Cambridge Phil. Soc. 93 (1983) 9–23.Google Scholar
  10. 10.
    H. Meijer, personal communication.Google Scholar
  11. 11.
    J. O'Rourke, Art Gallery Theorems and Algorithms, Oxford University Press, N.Y., 1987.Google Scholar
  12. 12.
    G. Ringel, Färbungsproblems auf Flächen und Graphen, Deutscher Verlag der Wissenschaften, Berlin, 1959.Google Scholar
  13. 13.
    E. Steinitz and H. Rademacher, Vorlesungen über die Theorie der Polyeder, Springer, Berlin, 1934.Google Scholar
  14. 14.
    R. Tamassia and I. G. Tollis, A unified approach to visibility representations of planar graphs, Disc. and Comp. Geom. 1 (1986) 321–341.CrossRefGoogle Scholar
  15. 15.
    C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory 12 (1988) 335–341.Google Scholar
  16. 16.
    W. T. Tutte, On the non-biplanar character of the complete 9-graph, Canad. Math. Bull. 6 (1963) 319–330.Google Scholar
  17. 17.
    -, The thickness of a graph, Indag. Math. 25 (1963) 567–577.Google Scholar
  18. 18.
    J. D. Ullman, Computational Aspects of VLSI Design, Computer Science Press, Rockville, Md., 1984.Google Scholar
  19. 19.
    S. K. Wismath, Characterizing bar line-of-sight graphs, Proc. 1st Symp. Comp. Geom., ACM (1985) 147–152.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Joan P. Hutchinson
    • 1
  • Thomas Shermer
    • 2
  • Andrew Vince
    • 3
  1. 1.Department of MathematicsMacalester CollegeSt. PaulUSA
  2. 2.Department of Computer ScienceSimon Fraser UniversityBurnabyCanada
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA

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