Characterizing Planarity by the Splittable Deque

Auer, Christopher and Brandenburg, Franz J. and Gleißner, Andreas and Hanauer, Kathrin (2013) Characterizing Planarity by the Splittable Deque. In: 21st International Symposium, GD 2013, September 23-25, 2013, Bordeaux, France , pp. 25-36 (Official URL:

Full text not available from this repository.


A graph layout describes the processing of a graph G by a data structure , and the graph is called a  -graph. The vertices of G are totally ordered in a linear layout and the edges are stored and organized in  . 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.

Item Type:Conference Paper
Classifications:G Algorithms and Complexity > G.490 Embeddings
G Algorithms and Complexity > G.770 Planarity Testing
ID Code:1358

Repository Staff Only: item control page


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)

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)

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)

Bernhart, F., Kainen, P.: The book thickness of a graph. J. Combin. Theory, Ser. B 27(3), 320–331 (1979)

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)

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)

Dujmović, V., Wood, D.R.: On linear layouts of graphs. Discrete Math. Theor. Comput. Sci. 6(2), 339–358 (2004)

Dujmović, V., Wood, D.R.: Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci. 7(1), 155–202 (2005)

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)

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)

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)

Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21(5), 927–958 (1992)

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)

Rosenstiehl, P., Tarjan, R.E.: Gauss codes, planar hamiltonian graphs, and stack-sortable permutations. J. of Algorithms 5, 375–390 (1984)

Shih, W.K., Hsu, W.L.: A new planarity test. Theor. Comput. Sci. 223(1-2), 179–191 (1999)

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)

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)

Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. 38(1), 36–67 (1989)