Characterizing Planarity by the Splittable Deque

  • Christopher Auer
  • Franz J. Brandenburg
  • Andreas Gleißner
  • Kathrin Hanauer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)


A graph layout describes the processing of a graph G by a data structure \(\mathcal{D}\), and the graph is called a \(\mathcal{D}\)-graph. The vertices of G are totally ordered in a linear layout and the edges are stored and organized in \(\mathcal{D}\). At each vertex, all edges to predecessors in the linear layout are removed and all edges to successors are inserted. There are intriguing relationships between well-known data structures and classes of planar graphs: The stack graphs are the outerplanar graphs [4], the queue graphs are the arched leveled-planar graphs [12], the 2-stack graphs are the subgraphs of planar graphs with a Hamilton cycle [4], and the deque graphs are the subgraphs of planar graphs with a Hamilton path [2]. All of these are proper subclasses of the planar graphs, even for maximal planar graphs.

We introduce splittable deques as a data structure to capture planarity. A splittable deque is a deque which can be split into sub-deques. The splittable deque provides a new insight into planarity testing by a game on switching trains. Here, we use it for a linear-time planarity test of a given rotation system.


  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.
    Auer, C., Gleißner, A.: Characterizations of deque and queue graphs. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 35–46. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Auer, C., Gleißner, A., Hanauer, K., Vetter, S.: Testing planarity by switching trains. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 557–558. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Bernhart, F., Kainen, P.: The book thickness of a graph. J. Combin. Theory, Ser. B 27(3), 320–331 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Donafee, A., Maple, C.: Planarity testing for graphs represented by a rotation scheme. In: Banissi, E., Börner, K., Chen, C., Clapworthy, G., Maple, C., Lobben, A., Moore, C.J., Roberts, J.C., Ursyn, A., Zhang, J. (eds.) Proc. Seventh International Conference on Information Visualization, IV 2003, pp. 491–497. IEEE Computer Society, Washington, DC (2003)Google Scholar
  7. 7.
    Dujmović, V., Wood, D.R.: On linear layouts of graphs. Discrete Math. Theor. Comput. Sci. 6(2), 339–358 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dujmović, V., Wood, D.R.: Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci. 7(1), 155–202 (2005)MathSciNetGoogle Scholar
  9. 9.
    de Fraysseix, H., Rosenstiehl, P.: A depth-first-search characterization of planarity. In: Graph Theory, Cambridge (1981); Ann. Discrete Math., vol. 13, pp. 75–80. North-Holland, Amsterdam (1982)Google Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21(5), 927–958 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kosaraju, S.R.: Real-time simulation of concatenable double-ended queues by double-ended queues (preliminary version). In: Proc. 11th Annual ACM Symposium on Theory of Computing, STOC 1979, pp. 346–351. ACM, New York (1979)Google Scholar
  14. 14.
    Rosenstiehl, P., Tarjan, R.E.: Gauss codes, planar hamiltonian graphs, and stack-sortable permutations. J. of Algorithms 5, 375–390 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shih, W.K., Hsu, W.L.: A new planarity test. Theor. Comput. Sci. 223(1-2), 179–191 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    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
  18. 18.
    Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Christopher Auer
    • 1
  • Franz J. Brandenburg
    • 1
  • Andreas Gleißner
    • 1
  • Kathrin Hanauer
    • 1
  1. 1.University of PassauGermany

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