Track Layout Is Hard

  • Michael J. Bannister
  • William E. DevannyEmail author
  • Vida Dujmović
  • David Eppstein
  • David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


We show that testing whether a given graph has a 3-track layout is hard, by characterizing the bipartite 3-track graphs in terms of leveled planarity. Additionally, we investigate the parameterized complexity of track layouts, showing that past methods used for book layouts do not work to parameterize the problem by treewidth or almost-tree number but that the problem is (non-uniformly) fixed-parameter tractable for tree-depth. We also provide several natural classes of bipartite planar graphs, including the bipartite outerplanar graphs, squaregraphs, and dual graphs of arrangements of monotone curves, that always have 3-track layouts.


  1. 1.
    Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM J. Comput. 34, 553–579 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dujmović, V., Wood, D.R.: Three-dimensional grid drawings with sub-quadratic volume. In: Pach, J., ed.: Towards a Theory of Geometric Graphs, vol. 342, pp. 55–66. Contemporary Mathematics - American Mathematical Society (2004)Google Scholar
  3. 3.
    Dujmović, V., Pór, A., Wood, D.R.: Track layouts of graphs. Discrete Math. Theor. Comput. Sci. 6, 497–521 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21, 927–958 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bannister, M.J., Eppstein, D.: Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 210–221. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-45803-7_18 Google Scholar
  6. 6.
    Bannister, M.J., Eppstein, D., Simons, J.A.: Fixed parameter tractability of crossing minimization of almost-trees. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 340–351. Springer, Heidelberg (2013). doi: 10.1007/978-3-319-03841-4_30 CrossRefGoogle Scholar
  7. 7.
    Harary, F., Schwenk, A.: A new crossing number for bipartite graphs. Utilitas Math. 1, 203–209 (1972)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Halin, R.: \(S\)-functions for graphs. J. Geom. 8, 171–186 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dujmović, V., Morin, P., Wood, D.R.: Layered separators for queue layouts, 3D graph drawing and nonrepetitive coloring. In: Proceedings of 54th Symposium on Foundations of Computer Science (FOCS 2013), pp. 280–289. IEEE Computer Society (2013)Google Scholar
  11. 11.
    Dujmović, V., Morin, P., Wood, D.R.: Layered separators in minor-closed graph classes with applications. Electronic preprint arXiv:1306.1595 (2013)
  12. 12.
    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybern. SMC–11, 109–125 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11, 379–403 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Healy, P., Nikolov, N.S.: How to layer a directed acyclic graph. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 16–30. Springer, Heidelberg (2002). doi: 10.1007/3-540-45848-4_2 CrossRefGoogle Scholar
  15. 15.
    Felsner, S., Liotta, G., Wismath, S.K.: Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithms Appl. 7, 363–398 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dujmović, V., Wood, D.R.: Stacks, queues and tracks: layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci. 7, 155–201 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Bannister, M.J., Cabello, S., Eppstein, D.: Parameterized complexity of 1-planarity. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 97–108. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40104-6_9 CrossRefGoogle Scholar
  19. 19.
    Nešetřil, J., de Mendez, P.O.: Sparsity (Graphs, Structures, and Algorithms). Algorithms and Combinatorics, vol. 28. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  20. 20.
    Abel, Z., Demaine, E.D., Demaine, M.L., Eppstein, D., Lubiw, A., Uehara, R.: Flat foldings of plane graphs with prescribed angles and edge lengths. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 272–283. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-45803-7_23 Google Scholar
  21. 21.
    Bandelt, H.J., Chepoi, V., Eppstein, D.: Combinatorics and geometry of finite and infinite squaregraphs. SIAM J. Discrete Math. 24, 1399–1440 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Michael J. Bannister
    • 1
  • William E. Devanny
    • 2
    Email author
  • Vida Dujmović
    • 3
  • David Eppstein
    • 2
  • David R. Wood
    • 4
  1. 1.Department of Mathematics and Computer ScienceSanta Clara UniversitySanta ClaraUSA
  2. 2.Department of Computer ScienceUniversity of California, IrvineIrvineUSA
  3. 3.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  4. 4.School of Mathematical SciencesMonash UniversityMelbourneAustralia

Personalised recommendations