Advertisement

Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing

  • Vladimir P. Korzhik
  • Bojan Mohar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G − e is 1-planar for every edge e of G. We construct two infinite families of minimal non-1-planar graphs and show that for every integer n ≥ 63, there are at least \(2^{\frac{n}{4}-\frac{54}{4}}\) non-isomorphic minimal non-1-planar graphs of order n. It is also proved that testing 1-planarity is NP-complete. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.

References

  1. 1.
    Borodin, O.V.: Solution of Ringel’s problem about vertex bound colouring of planar graphs and colouring of 1-planar graphs [in Russian]. Metody Discret. Analiz. 41, 12–26 (1984)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Borodin, O.V.: A new proof of the 6-color theorem. J. Graph Theory 19, 507–521 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Borodin, O.V., Kostochka, A.V., Raspaud, A., Sopena, E.: Acyclic colouring of 1-planar graphs. Discrete Analysis and Operations Researcher 6, 20–35 (1999)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Chen, Z.-Z.: Approximation algorithms for independent sets in map graphs. Journal of Algorithms 41, 20–40 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Z.-Z.: New bounds on the number of edges in a k-map graph. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 319–328. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Chen, Z.-Z., Kouno M.: A linear-time algorithm for 7-coloring 1-plane graphs. Algorithmica 43, 147–177 (2005)Google Scholar
  7. 7.
    Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Math. 307, 854–865 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comp. Sci. 1, 237–267 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hliněný, P.: Crossing number is hard for cubic graphs. J. Combin. Theory, Ser. B 96, 455–471 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Korzhik, V.P.: Minimal non-1-planar graphs. Discrete Math. 308, 1319–1327 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abh. Sem. Univ. Hamburg 29, 107–117 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Schumacher, H.: Zur Struktur 1-planarer Graphen. Math. Nachr. 125, 291–300 (1986)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Vladimir P. Korzhik
    • 1
  • Bojan Mohar
    • 2
  1. 1.National Academy of ScienceNational University of Chernivtsi and Institute of APMMLvivUkraine
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada

Personalised recommendations