Obtaining a Planar Graph by Vertex Deletion

  • Dániel Marx
  • Ildikó Schlotter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4769)

Abstract

In the Planar + k vertex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour [19,18], there is an O(n 3) time algorithm for every fixed value of k. However, the proof is extremely complicated and the constants hidden by the big-O notation are huge. Here we give a much simpler algorithm for this problem with quadratic running time, by iteratively reducing the input graph and then applying techniques for graphs of bounded treewidth.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)MATHCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inform. Process. Lett. 58, 171–176 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Courcelle, B.: Graph rewriting: an algebraic and logic approach. In: Handbook of Theoretical Computer Science, vol. 2, pp. 194–242. Elsevier, Amsterdam (1990)Google Scholar
  5. 5.
    Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Handbook of graph grammars and computing by graph transformations, ch. 5, vol. 1, pp. 313–400. World Scientific, New-Jersey (1997)Google Scholar
  6. 6.
    Diestel, R.: Graph Theory. Springer, Berlin (2000)Google Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)Google Scholar
  8. 8.
    Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3, 1–27 (1999)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. J. ACM 49, 716–752 (2002)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Crossing Number is NP-Complete. SIAM J. Algebraic Discrete Methods 4, 312–316 (1983)MATHMathSciNetGoogle Scholar
  11. 11.
    Grohe, M.: Computing crossing number in quadratic time. J. Comput. System Sci. 68, 285–302 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hadlock, F.: Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput. 4, 221–225 (1975)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hopcroft, J.E., Tarjan, R.E.: Efficient planarity testing. J. ACM 21, 549–568 (1974)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lewis, J.M., Yannakakis, M.: The vertex-deletion problem for hereditary properties is NP-complete. J. Comput. System Sci. 20, 219–230 (1980)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9, 615–627 (1980)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Perkovic, L., Reed, B.: An improved algorithm for finding tree decompositions of small width. Internat. J. Found. Comput. Sci. 11, 365–371 (2000)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Combin. Theory Ser. B 62, 323–348 (1994)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Robertson, N., Seymour, P.D.: Graph minors. J. Combin. Theory Ser. B 92, 325–357 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63, 65–110 (1995)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Thomassen, C.: The graph genus problem is NP-complete. J. Algorithms 10, 568–576 (1989)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Dániel Marx
    • 1
  • Ildikó Schlotter
    • 2
  1. 1.Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, BerlinGermany
  2. 2.Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest, H-1521Hungary

Personalised recommendations