Skip to main content
SpringerLink
Log in

The Local Queue Number of Graphs with Bounded Treewidth

  • Conference paper
  • First Online:
Graph Drawing and Network Visualization (GD 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12590))

Included in the following conference series:

Abstract

A queue layout of a graph G consists of a vertex ordering of G and a partition of the edges into so-called queues such that no two edges in the same queue nest, i.e., have their endpoints ordered in an ABBA-pattern. Continuing the research on local ordered covering numbers, we introduce the local queue number of a graph G as the minimum \(\ell \) such that G admits a queue layout with each vertex having incident edges in no more than \(\ell \) queues. Similarly to the local page number [Merker, Ueckerdt, GD’19], the local queue number is closely related to the graph’s density and can be arbitrarily far from the classical queue number.

We present tools to bound the local queue number of graphs from above and below, focusing on graphs of treewidth k. Using these, we show that every graph of treewidth k has local queue number at most \(k+1\) and that this bound is tight for \(k=2\), while a general lower bound is \(\lceil k/2\rceil +1\). Our results imply, inter alia, that the maximum local queue number among planar graphs is either 3 or 4.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aggarwal, A., Klawe, M., Shor, P.: Multilayer grid embeddings for VLSI. Algorithmica 6(1), 129–151 (1991). https://doi.org/10.1007/BF01759038

    Article  MathSciNet  MATH  Google Scholar 

  2. Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs IV: linear arboricity. Networks 11(1), 69–72 (1981). https://doi.org/10.1002/net.3230110108

    Article  MathSciNet  MATH  Google Scholar 

  3. Akiyama, J., Kano, M.: Path factors of a graph. Graph theory and its Applications, pp. 11–22 (1984)

    Google Scholar 

  4. Alam, J.M., Bekos, M.A., Gronemann, M., Kaufmann, M., Pupyrev, S.: Queue layouts of planar 3-trees. Algorithmica 82(9), 2564–2585 (2020). https://doi.org/10.1007/s00453-020-00697-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Comb. Theory Ser. B 27(3), 320–331 (1979). https://doi.org/10.1016/0095-8956(79)90021-2

    Article  MathSciNet  MATH  Google Scholar 

  6. Clote, P., Dobrev, S., Dotu, I., Kranakis, E., Krizanc, D., Urrutia, J.: On the page number of RNA secondary structures with pseudoknots. J. Math. Biol. 65(6), 1337–1357 (2012). https://doi.org/10.1007/s00285-011-0493-6

    Article  MathSciNet  MATH  Google Scholar 

  7. Dujmović, V., Joret, G., Micek, P., Morin, P., Ueckerdt, T., Wood, D.R.: Planar graphs have bounded queue-number. In: 60th IEEE Annual Symposium on Foundations of Computer Science (2019)

    Google Scholar 

  8. Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM J. Comput. 34(3), 553–579 (2005). https://doi.org/10.1137/S0097539702416141

    Article  MathSciNet  MATH  Google Scholar 

  9. Fishburn, P.C., Hammer, P.L.: Bipartite dimensions and bipartite degrees of graphs. Discrete Math. 160(1), 127–148 (1996). https://doi.org/10.1016/0012-365X(95)00154-O

  10. Gonçalves, D.: Caterpillar arboricity of planar graphs. Discrete Math. 307(16), 2112–2121 (2007). https://doi.org/10.1016/j.disc.2005.12.055. euroComb ’03 - Graphs and Algorithms

  11. Guy, R.K., Nowakowski, R.J.: The outerthickness & outercoarseness of graphs I. The complete graph & the \(n\)-cube. In: Bodendiek, R., Henn, R. (eds.) Topics in Combinatorics and Graph Theory: Essays in Honour of Gerhard Ringel, pp. 297–310. Physica-Verlag HD, Heidelberg (1990). https://doi.org/10.1007/978-3-642-46908-4_34

  12. Heath, L.S., Ribbens, C.J., Pemmaraju, S.V.: Processor-efficient sparse matrix-vector multiplication. Comput. Math. Appl. 48(3), 589–608 (2004). https://doi.org/10.1016/j.camwa.2003.06.009

    Article  MathSciNet  MATH  Google Scholar 

  13. Heath, L.S., Leighton, F., Rosenberg, A.: Comparing queues and stacks as machines for laying out graphs. SIAM J. Discrete Math. 5(3), 398–412 (1992). https://doi.org/10.1137/0405031

    Article  MathSciNet  MATH  Google Scholar 

  14. Heath, L.S., Rosenberg, A.: Laying out graphs using queues. SIAM J. Comput. 21(5), 927–958 (1992). https://doi.org/10.1137/0221055

    Article  MathSciNet  MATH  Google Scholar 

  15. Joseph, D., Meidanis, J., Tiwari, P.: Determining DNA sequence similarity using maximum independent set algorithms for interval graphs. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 326–337. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55706-7_29

    Chapter  Google Scholar 

  16. Knauer, K., Ueckerdt, T.: Three ways to cover a graph. Discrete Math. 339(2), 745–758 (2016). https://doi.org/10.1016/j.disc.2015.10.023

    Article  MathSciNet  MATH  Google Scholar 

  17. Merker, L., Ueckerdt, T.: Local and union page numbers. In: Archambault, D., Tóth, C.D. (eds.) GD 2019. LNCS, vol. 11904, pp. 447–459. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-35802-0_34

    Chapter  MATH  Google Scholar 

  18. Merker, L., Ueckerdt, T.: The local queue number of graphs with bounded treewidth (2020). https://arxiv.org/abs/2008.05392

  19. Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Comb. 14(1), 59–73 (1998). https://doi.org/10.1007/PL00007219

    Article  MathSciNet  MATH  Google Scholar 

  20. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. J. Lond. Math. Soc. s1–39(1), 12–12 (1964). https://doi.org/10.1112/jlms/s1-39.1.12

  21. Ramanathan, S., Lloyd, E.L.: Scheduling algorithms for multi-hop radio networks. SIGCOMM Comput. Commun. Rev. 22(4), 211–222 (1992). https://doi.org/10.1145/144191.144283

    Article  Google Scholar 

  22. 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). https://doi.org/10.1007/BFb0030834

    Chapter  Google Scholar 

  23. Rosenberg, A.: The diogenes approach to testable fault-tolerant arrays of processors. IEEE Trans. Comput. C-32(10), 902–910 (1983). https://doi.org/10.1109/TC.1983.1676134

  24. Skums, P.V., Suzdal, S.V., Tyshkevich, R.I.: Edge intersection graphs of linear 3-uniform hypergraphs. Discrete Math. 309(11), 3500–3517 (2009). https://doi.org/10.1016/j.disc.2007.12.082. 7th International Colloquium on Graph Theory

  25. Wiechert, V.: On the queue-number of graphs with bounded tree-width. Electr. J. Comb. 24(1), P1.65 (2017). http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p65

  26. 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). https://doi.org/10.1007/3-540-36206-1_31

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Theoretical Informatics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Laura Merker & Torsten Ueckerdt

Authors
  1. Laura Merker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Torsten Ueckerdt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laura Merker .

Editor information

Editors and Affiliations

  1. LaBRI, University of Bordeaux, Talence, France

    David Auber

  2. Charles University, Prague, Czech Republic

    Pavel Valtr

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Merker, L., Ueckerdt, T. (2020). The Local Queue Number of Graphs with Bounded Treewidth. In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-68766-3_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-68765-6

  • Online ISBN: 978-3-030-68766-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation