On the Page Number of Upward Planar Directed Acyclic Graphs

  • Fabrizio Frati
  • Radoslav Fulek
  • Andres J. Ruiz-Vargas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)

Abstract

In this paper we study the page number of upward planar directed acyclic graphs. We prove that: (1) the page number of any n-vertex upward planar triangulation G whose every maximal 4-connected component has page number k is at most min {O(klogn),O(2 k )}; (2) every upward planar triangulation G with \(o(\frac{n}{\log n})\) diameter has o(n) page number; and (3) every upward planar triangulation has a vertex ordering with o(n) page number if and only if every upward planar triangulation whose maximum degree is \(O(\sqrt n)\) does.

References

  1. 1.
    Alzohairi, M., Rival, I.: Series-Parallel Planar Ordered Sets Have Pagenumber Two. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 11–24. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  2. 2.
    Buss, J.F., Shor, P.W.: On the pagenumber of planar graphs. In: Symposium on Theory of Computing (STOC 1984), pp. 98–100. ACM (1984)Google Scholar
  3. 3.
    Cerný, J.: Coloring circle graphs. Elec. Notes Discr. Math. 29, 457–461 (2007)CrossRefMATHGoogle Scholar
  4. 4.
    Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: A layout problem with applications to VLSI design. SIAM J. Alg. Discr. Meth. 8, 33–58 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theor. Comp. Sci. 61, 175–198 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Book embeddability of series-parallel digraphs. Algorithmica 45(4), 531–547 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Enomoto, H., Nakamigawa, T., Ota, K.: On the pagenumber of complete bipartite graphs. J. Comb. Th. Ser. B 71(1), 111–120 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ganley, J.L., Heath, L.S.: The pagenumber of k-trees is O(k). Discr. Appl. Math. 109(3), 215–221 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Heath, L.S.: Embedding planar graphs in seven pages. In: Foundations of Computer Science (FOCS 1984), pp. 74–83. IEEE (1984)Google Scholar
  10. 10.
    Heath, L.S., Istrail, S.: The pagenumber of genus g graphs is O(g). J. ACM 39(3), 479–501 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Heath, L.S., Leighton, F.T., Rosenberg, A.L.: Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discr. Math. 5(3), 398–412 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Heath, L.S., Pemmaraju, S.V.: Stack and queue layouts of posets. SIAM J. Discr. Math. 10(4), 599–625 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Heath, L.S., Pemmaraju, S.V.: Stack and queue layouts of directed acyclic graphs: Part II. SIAM J. Computing 28(5), 1588–1626 (1999)CrossRefMATHGoogle Scholar
  14. 14.
    Heath, L.S., Pemmaraju, S.V., Trenk, A.N.: Stack and queue layouts of directed acyclic graphs: Part I. SIAM J. Computing 28(4), 1510–1539 (1999)CrossRefMATHGoogle Scholar
  15. 15.
    Kainen, P.C.: Thickness and coarseness of graphs. Abh. Math. Sem. Univ. Hamburg 39, 88–95 (1973)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kostochka, A.V., Kratochvíl, J.: Covering and coloring polygon-circle graphs. Discr. Math. 163(1-3), 299–305 (1997)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Malitz, S.M.: Genus g graphs have pagenumber \({O}(\sqrt g)\). J. Algorithms 17(1), 85–109 (1994)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Malitz, S.M.: Graphs with e edges have pagenumber \({O}(\sqrt e)\). J. Algorithms 17(1), 71–84 (1994)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ollmann, L.T.: On the book thicknesses of various graphs. In: Hoffman, F., Levow, R.B., Thomas, R.S.D. (eds.) Southeastern Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium, vol. VIII, p. 459 (1973)Google Scholar
  20. 20.
    Rosenberg: The Diogenes approach to testable fault-tolerant arrays of processors. IEEE Trans. Comp. C-32, 902–910 (1983)CrossRefGoogle Scholar
  21. 21.
    Tarjan, R.E.: Sorting using networks of queues and stacks. J. ACM 19(2), 341–346 (1972)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Yannakakis, M.: Embedding planar graphs in four pages. J. Comp. Syst. Sci. 38(1), 36–67 (1989)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Fabrizio Frati
    • 1
    • 2
    • 3
  • Radoslav Fulek
    • 1
  • Andres J. Ruiz-Vargas
    • 1
  1. 1.School of Basic SciencesÉcole Polytechnique Fédérale de LausanneSwitzerland
  2. 2.Dipartimento di Informatica e AutomazioneUniversità Roma TreItaly
  3. 3.School of Information TechnologiesUniversity of SydneyAustralia

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