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Maximum Order of Planar Digraphs

  • Rinovia Simanjuntak
  • Mirka Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3330)

Abstract

We consider the degree/diameter problem for directed planar graphs. We show that planar digraphs with diameter 2 and maximum out-degree and in-degree d, d ≥ 41, cannot have more than 2d vertices. We show that 2d is the best possible upper bound by constructing planar digraphs of diameter 2 having exactly 2d vertices.

Furthermore, we give upper and lower bounds for the largest possible order of planar digraphs with diameter greater than 2.

Keywords

Planar Graph Maximum Degree Directed Path Regular Graph Maximum Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bannai, E., Ito, T.: On finite Moore graphs. J. Fac. Sci. Tokyo Univ. 20, 191–208 (1973)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bannai, E., Ito, T.: Regular graphs with excess one. Discrete Math. 37, 147–158 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baskoro, E.T., Miller, M., Širáň, J., Sutton, M.: A complete characterisation of almost Moore digraphs of degree three. J. Graph theory (in press)Google Scholar
  4. 4.
    Bridges, W.G., Toueg, S.: On the impossibility of directed Moore graphs. J. Combin. Th. 29, 339–341 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Damerell, R.M.: On Moore graphs. Proc. Cambridge Philos. Soc. 74, 227–236 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fellows, M., Hell, P., Seyffarth, K.: Constructions of dense planar networks, Technical Report. DCS-210-IR, University of Victoria (1993)Google Scholar
  7. 7.
    Fellows, M., Hell, P., Seyffarth, K.: Large planar graphs with given diameter and maximum degree. Discrete App. Math. 61, 133–153 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fellows, M., Hell, P., Seyffarth, K.: Construction of large planar networks with given degree and diameter. Networks 32, 275–281 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Göbel, F., Kern, W.: Planar regular graphs with prescribed diameter. Univ. of Twente (The Netherlands) Applied Math. Memorandum 1183 (1993)Google Scholar
  10. 10.
    Hell, P., Seyffarth, K.: Largest planar graphs of diameter two and fixed maximum degree. Discrete Math. 111, 313–322 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hoffman, A.J., Singleton, R.R.: On Moore graphs with diameter 2 and 3. IBM J. Res. Develop. 4, 497–504 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Jorgensen, L.K.: Diameters of cubic graphs. Discrete App. Math. 37/38, 347–351 (1992)CrossRefGoogle Scholar
  13. 13.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math 36, 177–189 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Miller, M., Fris, I.: Maximum order digraphs for diameter 2 or degree 2. In: Pullman vol. of Graphs and Matrices. Lecture Notes in Pure and App. Math., vol. 139, pp. 269–278Google Scholar
  15. 15.
    Miller, M., Širáň, J.: Digraphs of degree two which miss the Moore bound by two. Discrete Math. 226, 269–280 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Plesník, J., Znám, Š.: Strongly geodetic directed graphs. Acta F. R. N. Univ. Comen. - Mathematica XXIX, 29–34 (1974)Google Scholar
  17. 17.
    Pratt, R.W.: The (Degree, Diameter) Problem for Planar Graphs, http://www.unc.edu/~rpratt/degdiam.html
  18. 18.
    Seyffarth, K.: Maximal planar graphs of diameter two. J. Graph Theory 13, 619–648 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Simanjuntak, R., Miller, M.: Largest planar digraphs of diameter 2. In: Proceedings of Thirteenth Australasian Workshop of Combinatorial Algorithm, Fraser Island, Australia, July 7-10, pp. 43–51 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Rinovia Simanjuntak
    • 1
  • Mirka Miller
    • 1
  1. 1.School of Electrical Engineering and Computer ScienceThe University of NewcastleAustralia

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