Parameterized Complexity of 1-Planarity

  • Michael J. Bannister
  • Sergio Cabello
  • David Eppstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

We consider the problem of finding a 1-planar drawing for a general graph, where a 1-planar drawing is a drawing in which each edge participates in at most one crossing. Since this problem is known to be NP-hard we investigate the parameterized complexity of the problem with respect to the vertex cover number, tree-depth, and cyclomatic number. For these parameters we construct fixed-parameter tractable algorithms. However, the problem remains NP-complete for graphs of bounded bandwidth, pathwidth, or treewidth.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akutsu, T., Hayashida, M., Ching, W.-K., Ng, M.K.: Control of Boolean networks: Hardness results and algorithms for tree structured networks. J. Theor. Biol. 244(4), 670–679 (2007), doi:10.1016/j.jtbi.2006.09.023MathSciNetCrossRefGoogle Scholar
  2. 2.
    Borodin, O.V.: Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs. Metody Diskret. Analiz. (41), 12–26, 108 (1984)Google Scholar
  3. 3.
    Brandenburg, F.J., Eppstein, D., Gleißner, A., Goodrich, M.T., Hanauer, K., Reislhuber, J.: On the density of maximal 1-planar graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 327–338. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. CoRR abs/1203.5944 (2012)Google Scholar
  5. 5.
    Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theoretical Computer Science 411(40-42), 3736–3756 (2010), doi:10.1016/j.tcs.2010.06.026MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Chen, Z.-Z., Kouno, M.: A linear-time algorithm for 7-coloring 1-plane graphs. Algorithmica 43(3), 147–177 (2005), doi:10.1007/s00453-004-1134-xMathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Czap, J., Hudák, D.: 1-planarity of complete multipartite graphs. Discrete Applied Mathematics 160(4-5), 505–512 (2012), doi:10.1016/j.dam.2011.11.014MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Damaschke, P.: Induced subgraphs and well-quasi-ordering. J. Graph Th. 14(4), 427–435 (1990), doi:10.1002/jgt.3190140406MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer (1999), doi:10.1007/978-1-4612-0515-9Google Scholar
  10. 10.
    Eades, P., Hong, S.-H., Katoh, N., Liotta, G., Schweitzer, P., Suzuki, Y.: Testing maximal 1-planarity of graphs with a rotation system in linear time. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 339–345. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Eades, P., Liotta, G.: Right angle crossing graphs and 1-planarity. In: Speckmann, B. (ed.) GD 2011. LNCS, vol. 7034, pp. 148–153. Springer, Heidelberg (2011)Google Scholar
  12. 12.
    Fernandez-Baca, D.: Allocating modules to processors in a distributed system. IEEE Transactions on Software Engineering 15(11), 1427–1436 (1989), doi:10.1109/32.41334CrossRefGoogle Scholar
  13. 13.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. Springer (2006)Google Scholar
  14. 14.
    Grigoriev, A., Bodlaender, H.L.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1–11 (2007), doi:10.1007/s00453-007-0010-xMathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of generalized vertex cover problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 36–48. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Gurevich, Y., Stockmeyer, L., Vishkin, U.: Solving NP-Hard Problems on Graphs That Are Almost Trees and an Application to Facility Location Problems. J. ACM 31(3), 459–473 (1984), doi:10.1145/828.322439MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hong, S.-H., Eades, P., Liotta, G., Poon, S.-H.: Fáry’s theorem for 1-planar graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 335–346. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Korzhik, V.P.: Minimal non-1-planar graphs. Discrete Mathematics 308(7), 1319–1327 (2008), doi:10.1016/j.disc.2007.04.009MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Korzhik, V.P., Mohar, B.: Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing. J. Graph Th. 72(1), 30–71 (2013), doi:10.1002/jgt.21630MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Miller, G.L.: Finding Small Simple Cycle Separators for 2-Connected Planar Graphs. J. Comput. Syst. Sci. 32(3), 265–279 (1986), doi:10.1016/0022-0000(86)90030-9MATHCrossRefGoogle Scholar
  21. 21.
    Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms. Algorithms and Combinatorics 28, 115–144 (2012), doi:10.1007/978-3-642-27875-4CrossRefGoogle Scholar
  22. 22.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997), doi:10.1007/BF01215922MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 29, 107–117 (1965), doi:10.1007/BF02996313MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Schumacher, H.: Zur Struktur 1-planarer Graphen. Mathematische Nachrichten 125, 291–300 (1986)MathSciNetMATHGoogle Scholar
  25. 25.
    Seidman, S.B.: Network structure and minimum degree. Social Networks 5(3), 269–287 (1983), doi:10.1016/0378-8733(83)90028-XMathSciNetCrossRefGoogle Scholar
  26. 26.
    Suzuki, Y.: Optimal 1-planar graphs which triangulate other surfaces. Discrete Mathematics 310(1), 6–11 (2010), doi:10.1016/j.disc.2009.07.016MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael J. Bannister
    • 1
  • Sergio Cabello
    • 2
  • David Eppstein
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaSlovenia

Personalised recommendations