Strip Planarity Testing

  • Patrizio Angelini
  • Giordano Da Lozzo
  • Giuseppe Di Battista
  • Fabrizio Frati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

Abstract

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph G(V,E) and a function γ:V → {1,2,…,k} and asks whether a planar drawing of G exists such that each edge is monotone in the y-direction and, for any u,v ∈ V with γ(u) < γ(v), it holds y(u) < y(v). The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. We show that the problem is polynomial-time solvable if G has a fixed planar embedding.

References

  1. 1.
    Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: Charikar, M. (ed.) SODA 2010, pp. 202–221. ACM (2010)Google Scholar
  2. 2.
    Angelini, P., Frati, F., Kaufmann, M.: Straight-line rectangular drawings of clustered graphs. Discrete & Computational Geometry 45(1), 88–140 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F.: Strip planarity testing of embedded planar graphs. ArXiv e-prints 1309.0683 (September 2013)Google Scholar
  4. 4.
    Bachmaier, C., Brandenburg, F.J., Forster, M.: Radial level planarity testing and embedding in linear time. JGAA 9(1), 53–97 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bertolazzi, P., Di Battista, G., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Biedl, T.C., Kaufmann, M., Mutzel, P.: Drawing planar partitions II: HH-drawings. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 124–136. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: Clustering cycles into cycles of clusters. JGAA 9(3), 391–413 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: On embedding a cycle in a plane graph. Discrete Mathematics 309(7), 1856–1869 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Di Battista, G., Frati, F.: Efficient c-planarity testing for embedded flat clustered graphs with small faces. JGAA 13(3), 349–378 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61, 175–198 (1988)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eades, P., Feng, Q., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Estrella-Balderrama, A., Fowler, J.J., Kobourov, S.G.: On the characterization of level planar trees by minimal patterns. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 69–80. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Forster, M., Bachmaier, C.: Clustered level planarity. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 218–228. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Fowler, J.J., Kobourov, S.G.: Minimum level nonplanar patterns for trees. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 69–75. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gutwenger, C., Klein, K., Mutzel, P.: Planarity testing and optimal edge insertion with embedding constraints. JGAA 12(1), 73–95 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Healy, P., Kuusik, A., Leipert, S.: A characterization of level planar graphs. Discrete Mathematics 280(1-3), 51–63 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hong, S.H., Nagamochi, H.: Two-page book embedding and clustered graph planarity. Tech. Report 2009-004, Dept. of Applied Mathematics & Physics, Kyoto University (2009)Google Scholar
  20. 20.
    Hopcroft, J.E., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hutton, M.D., Lubiw, A.: Upward planarity testing of single-source acyclic digraphs. SIAM J. Comput. 25(2), 291–311 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jelínek, V., Kratochvíl, J., Rutter, I.: A kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. Theory Appl. 46(4), 466–492 (2013)CrossRefMATHGoogle Scholar
  23. 23.
    Jelínková, E., Kára, J., Kratochvíl, J., Pergel, M., Suchý, O., Vyskocil, T.: Clustered planarity: Small clusters in cycles and Eulerian graphs. JGAA 13(3), 379–422 (2009)CrossRefMATHGoogle Scholar
  24. 24.
    Jünger, M., Leipert, S., Mutzel, P.: Level planarity testing in linear time. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 224–237. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  25. 25.
    Schaefer, M.: Toward a theory of planarity: Hanani-tutte and planarity variants. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 162–173. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Giordano Da Lozzo
    • 1
  • Giuseppe Di Battista
    • 1
  • Fabrizio Frati
    • 2
  1. 1.Dipartimento di IngegneriaRoma Tre UniversityItaly
  2. 2.School of Information TechnologiesThe University of SydneyAustralia

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