Characterizations of Deque and Queue Graphs

  • Christopher Auer
  • Andreas Gleißner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)

Abstract

In graph layouts the vertices of a graph are processed according to a linear order and the edges correspond to items in a data structure inserted and removed at their end vertices. Graph layouts characterize interesting classes of planar graphs: A graph G is a stack graph if and only if G is outerplanar, and a graph is a 2-stack graph if and only if it is a subgraph of a planar graph with a Hamiltonian cycle [2]. Heath and Rosenberg [12] characterized all queue graphs as the arched leveled-planar graphs. In [1], we have introduced linear cylindric drawings (LCDs) to study graph layouts in the double-ended queue (deque) and have shown that G is a deque graph if and only if it permits a plane LCD.

In this paper, we show that a graph is a deque graph if and only if it is the subgraph of a planar graph with a Hamiltonian path. In consequence, we obtain that the dual of an embedded queue graph contains a Eulerian path. We also turn to the respective decision problem of deque graphs and show that it is \(\mathcal{NP}\)-hard by proving that the Hamiltonian path problem in maximal planar graphs is \(\mathcal{NP}\)-hard. Heath and Rosenberg state [12] that queue graphs are “almost” proper leveled-planar. We show that bipartiteness captures this “almost”: A graph is proper leveled-planar if and only if it is a bipartite queue graph.

Keywords

Planar Graph Hamiltonian Cycle Front Line Hamiltonian Path Outerplanar Graph 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Auer, C., Bachmaier, C., Brandenburg, F.J., Brunner, W., Gleißner, A.: Plane Drawings of Queue and Deque Graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 68–79. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Bernhart, F., Kainen, P.: The book thickness of a graph. J. Combin. Theory, Ser. B 27(3), 320–331 (1979)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: A layout problem with applications to VLSI design. SIAM J. Algebra. Discr. Meth. 8(1), 33–58 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chvátal, V.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley and Sons, New York (1985)Google Scholar
  5. 5.
    Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Transactions on Systems, Man, and Cybernetics 18(6), 1035–1046 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dujmović, V., Wood, D.R.: On linear layouts of graphs. Discrete Math. Theor. Comput. Sci. 6(2), 339–358 (2004)MathSciNetMATHGoogle Scholar
  7. 7.
    Dujmović, V., Wood, D.R.: Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci. 7(1), 155–202 (2005)MathSciNetMATHGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)MATHGoogle Scholar
  9. 9.
    Heath, L.S., Leighton, F.T., Rosenberg, A.L.: Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discret. Math. 5(3), 398–412 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Heath, L.S., Pemmaraju, S.V.: Stack and queue layouts of directed acyclic graphs: Part II. SIAM J. Comput. 28(5), 1588–1626 (1999)CrossRefMATHGoogle Scholar
  11. 11.
    Heath, L.S., Pemmaraju, S.V., Trenk, A.N.: Stack and queue layouts of directed acyclic graphs: Part I. SIAM J. Comput. 28(4), 1510–1539 (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21(5), 927–958 (1992)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Rosenstiehl, P., Tarjan, R.E.: Gauss codes, planar hamiltonian graphs, and stack-sortable permutations. J. of Algorithms 5, 375–390 (1984)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Transactions on Systems, Man, and Cybernetics 11(2), 109–125 (1981)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wiegers, M.: Recognizing Outerplanar Graphs in Linear Time. In: Tinhofer, G., Schmidt, G. (eds.) WG 1986. LNCS, vol. 246, pp. 165–176. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  16. 16.
    Wigderson, A.: The complexity of the Hamiltonian circuit problem for maximal planar graphs. Tech. rep., Department of EECS, Princeton University (1982)Google Scholar
  17. 17.
    Wood, D.R.: Bounded Degree Book Embeddings and Three-Dimensional Orthogonal Graph Drawing. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 312–327. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Wood, D.R.: Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 348–359. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Yannakakis, M.: Four pages are necessary and sufficient for planar graphs. In: Proc. of the 18th Annual ACM Symposium on Theory of Computing, STOC 1986, pp. 104–108. ACM, New York (1986)Google Scholar
  20. 20.
    Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Christopher Auer
    • 1
  • Andreas Gleißner
    • 1
  1. 1.University of PassauPassauGermany

Personalised recommendations