Queue Layouts of Planar 3-Trees

  • Jawaherul Md. Alam
  • Michael A. Bekos
  • Martin GronemannEmail author
  • Michael Kaufmann
  • Sergey Pupyrev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)


A queue layout of a graph G consists of a linear order of the vertices of G and a partition of the edges of G into queues, so that no two independent edges of the same queue are nested. The queue number of G is the minimum number of queues required by any queue layout of G. In this paper, we continue the study of the queue number of planar 3-trees. As opposed to general planar graphs, whose queue number is not known to be bounded by a constant, the queue number of planar 3-trees has been shown to be at most seven. In this work, we improve the upper bound to five. We also show that there exist planar 3-trees, whose queue number is at least four; this is the first example of a planar graph with queue number greater than three.


  1. 1.
    Alam, J.M., Bekos, M.A., Gronemann, M., Kaufmann, M., Pupyrev, S.: Queue layouts of planar 3-trees. CoRR abs/1808.10841 (2018)Google Scholar
  2. 2.
    Bhatt, S.N., Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Scheduling tree-dags using FIFO queues: a control-memory trade-off. J. Parallel Distrib. Comput. 33(1), 55–68 (1996)CrossRefGoogle Scholar
  3. 3.
    Di Battista, G., Frati, F., Pach, J.: On the queue number of planar graphs. SIAM J. Comput. 42(6), 2243–2285 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dujmović, V.: Graph layouts via layered separators. J. Comb. Theory Ser. B 110, 79–89 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dujmović, V., Frati, F.: Stack and queue layouts via layered separators. J. Graph Algorithms Appl. 22(1), 89–99 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM J. Comput. 34(3), 553–579 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dujmović, V., Pór, A., Wood, D.R.: Track layouts of graphs. Discret. Math. Theor. Comput. Sci. 6(2), 497–522 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dujmović, V., Wood, D.R.: Tree-partitions of k-trees with applications in graph layout. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 205–217. Springer, Heidelberg (2003). Scholar
  9. 9.
    Dujmović, V., Wood, D.R.: Stacks, queues and tracks: layouts of graph subdivisions. Discret. Math. Theor. Comput. Sci. 7(1), 155–202 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hasunuma, T.: Laying out iterated line digraphs using queues. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 202–213. Springer, Heidelberg (2004). Scholar
  11. 11.
    Heath, L.S., Leighton, F.T., Rosenberg, A.L.: Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discrete Math. 5(3), 398–412 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21(5), 927–958 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mondal, D., Nishat, R.I., Rahman, M.S., Alam, M.J.: Minimum-area drawings of plane 3-trees. J. Graph Algorithms Appl. 15(2), 177–204 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ollmann, T.: On the book thicknesses of various graphs. In: Hoffman, F., Levow, R., Thomas, R. (eds.) Southeastern Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium, vol. VIII, p. 459 (1973)Google Scholar
  15. 15.
    Pach, J., Thiele, T., Tóth, G.: Three-dimensional grid drawings of graphs. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 47–51. Springer, Heidelberg (1997). Scholar
  16. 16.
    Pemmaraju, S.V.: Exploring the powers of stacks and queues via graph layouts. Ph.D. thesis, Virginia Tech (1992)Google Scholar
  17. 17.
    Pupyrev, S.: Mixed linear layouts of planar graphs. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 197–209. Springer, Cham (2018). Scholar
  18. 18.
    Rengarajan, S., Veni Madhavan, C.E.: Stack and queue number of 2-trees. In: Du, D.-Z., Li, M. (eds.) COCOON 1995. LNCS, vol. 959, pp. 203–212. Springer, Heidelberg (1995). Scholar
  19. 19.
    Shahrokhi, F., Shi, W.: On crossing sets, disjoint sets, and pagenumber. J. Algorithms 34(1), 40–53 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tarjan, R.E.: Sorting using networks of queues and stacks. J. ACM 19(2), 341–346 (1972)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wiechert, V.: On the queue-number of graphs with bounded tree-width. Electr. J. Comb. 24(1) (2017). P1.65Google Scholar
  22. 22.
    Wood, D.R.: Queue layouts, tree-width, and three-dimensional graph drawing. In: Agrawal, M., Seth, A. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 348–359. Springer, Heidelberg (2002). Scholar
  23. 23.
    Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Jawaherul Md. Alam
    • 1
  • Michael A. Bekos
    • 2
  • Martin Gronemann
    • 3
    Email author
  • Michael Kaufmann
    • 2
  • Sergey Pupyrev
    • 1
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  2. 2.Institut für InformatikUniversität TübingenTübingenGermany
  3. 3.Institut für InformatikUniversität zu KölnKölnGermany

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