An Algorithm for Finding Large Induced Planar Subgraphs

  • Keith Edwards
  • Graham Farr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)


This paper presents an efficient algorithm that finds an induced planar subgraph of at least 3n/(d + 1) vertices in a graph of n vertices and maximum degree d. This bound is sharp for d = 3, in the sense that if ɛ > 3/4 then there are graphs of maximum degree 3 with no induced planar subgraph of at least ɛn vertices. Our performance ratios appear to be the best known for small d. For example, when d = 3, our performance ratio of at least 3/4 compares with the ratio 1/2 obtained by Halldórsson and Lau. Our algorithm builds up an induced planar subgraph by iteratively adding a new vertex to it, or swapping a vertex in it with one outside it, in such a way that the procedure is guaranteed to stop, and so as to preserve certain properties that allow its performance to be analysed. This work is related to the authors’ work on fragmentability of graphs.


  1. 1.
    S. Bühler, Planarity of Graphs, M.Sc. dissertation, University of Dundee, 2000.Google Scholar
  2. 2.
    J. Cai and X. Han and R. E. Tarjan, An O(mlog n)-Time Algorithm for the Maximal Planar Subgraph Problem, SIAM J. Comput. 22 (1993) 1142–1162.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Călinescu, C. G. Fernandes, U. Finkler and H. Karloff, A better approximation algorithm for finding planar subgraphs, in: Proc. 7th Ann. ACM-SIAM Symp. on Discrete Algorithms (Atlanta, GA, 1996); also: J. Algorithms 27 (1998) 269–302.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Robert Cimikowski, An analysis of some heuristics for the maximum planar subgraph problem, in: Proc. 6th Ann. ACM-SIAM Symp. on Discrete Algorithms (San Francisco, CA, 1995), pp 322–331, ACM, New York, 1995.Google Scholar
  5. 5.
    G. Di Battista, Peter Eades, Roberto Tamassia and I. Tollis, Graph Drawing:A lgorithms for the Visualization of Graphs, Prentice Hall, 1999.Google Scholar
  6. 6.
    H. N. Djidjev and S. M. Venkatesan, Planarization of graphs embedded on surfaces, in: M. Nagl (ed.), Proc. 21st Internat. Workshop on Graph-Theoretic Concepts in Computer Science (WG’95) (Aachen, 1995), Lecture Notes in Comput. Sci., 1017, Springer, Berlin, 1995, pp. 62–72.Google Scholar
  7. 7.
    M. E. Dyer, L. R. Foulds and A. M. Frieze, Analysis of heuristics for finding a maximum weight planar subgraph, Eur. J. Operational Res. 20 (1985) 102–114.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    K. Edwards and G. Farr, Fragmentability of graphs, J. Combin. Theory (Ser. B) 82 (2001) 30–37.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    L. Faria, C. M. H. de Figueiredo and C. F. X. Mendonça, Splitting number is NP-complete, in: Juraj Hromkovic and Ondrej Sýkora (eds.), Proc. 24th Internat. Workshop on Graph-Theoretic Concepts in Computer Science (WG’ 98) (Smolenice Castle, Slovakia, 18-20 June 1998), Lecture Notes in Comput. Sci., 1517, Springer, 1998, pp. 285–297.Google Scholar
  10. 10.
    O. Goldschmidt and A. Takvorian, An efficient graph planarization two-phase heuristic, Networks 24 (1994) 69–73.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Magnús M. Halldórsson, Approximation of weighted independent set and hereditary subset problems, Journal of Graph Algorithms and Applications 4 (1) (2000) 1–16.MathSciNetGoogle Scholar
  12. 12.
    Magnús M. Halldórsson and Hoong Chuin Lau, Low-degree graph partitioning via local search with applications to constraint satisfaction, max cut, and coloring, Journal of Graph Algorithms and Applications 1 (3) (1997) 1–13.MathSciNetGoogle Scholar
  13. 13.
    J. P. Hutchinson, On genus-reducing and planarizing algorithms for embedded graphs, in: R. B. Richter (ed.), Graphs and Algorithms, Contemporary Mathematics 89, Amer. Math. Soc., Providence, RI, 1989.Google Scholar
  14. 14.
    Joan P. Hutchinson and Gary L. Miller, On deleting vertices to make a graph of positive genus planar, Discrete algorithms and complexity (Kyoto, 1986), Perspect. Comput., 15, pp. 81–98, Academic Press, Boston, MA, 1987.Google Scholar
  15. 15.
    M. Jünger and P. Mutzel, The polyhedral approach to the maximum planar subgraph problem: new chances for related problems, Graph Drawing’ 94, pp. 119–130.Google Scholar
  16. 16.
    M. Jünger and P. Mutzel, Maximum planar subgraphs and nice embeddings: practical layout tools, Algorithmica 16 (1996) 33–59.MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    P. C. Kainen, A generalization of the 5-color theorem, Proc. Amer. Math. Soc. 45 (1974) 450–453.Google Scholar
  18. 18.
    M. S. Krishnamoorthy and N. Deo, Node-deletion NP-complete problems, SIAM J. Comput. 8 (1979) 619–625.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    J. M. Lewis and M. Yannakakis, The node-deletion problem for hereditary properties is NP-complete, J. Comput. System Sci. 20 (1980) 219–230.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Annegret Liebers, Planarizing graphs — a survey and annotated bibliography, Journal of Graph Algorithms and Applications 5 (1) (2001) 1–74.MathSciNetGoogle Scholar
  21. 21.
    P. C. Liu and R. C. Geldmacher, On the deletion of nonplanar edges of a graph, Proc. 10th. S-E Conf. on Comb., Graph Theory, and Comp. 1977, Congressus Numerantium, No. 24 (1979) 727–738.Google Scholar
  22. 22.
    C. Lund and M. Yannakakis, The approximation of maximum subgraph problems, in: Proc. 20th Int. Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Comput. Sci., 700, Springer-Verlag, 1993, pp. 40–51.Google Scholar
  23. 23.
    P. Mutzel, The Maximum Planar Subgraph Problem, Ph.D. Thesis, Univ. zu Köln, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Keith Edwards
    • 1
  • Graham Farr
    • 2
  1. 1.Department of Applied ComputingUniversity of DundeeDundeeUK
  2. 2.School of Computer Science and Software EngineeringMonash University (Clayton Campus)ClaytonAustralia

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