Optimization and Recognition for K5-minor Free Graphs in Linear Time

  • Bruce Reed
  • Zhentao Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)


We present a linear time algorithm which determines whether an input graph contains K 5 as a minor and outputs a K 5-model if the input graph contains one. If the input graph has no K 5-minor then the algorithm constructs a tree decomposition such that each node of the tree corresponds to a planar graph or a graph with eight vertices. Such a decomposition can be used to obtain algorithms to solve various optimization problems in linear time. For example, we present a linear time algorithm for finding an \(O(\sqrt{n})\) seperator and a linear time algorithm for solving k-realisation on graphs without a K 5-minor. Our algorithm will also be used, in a separate paper, as a key subroutine in a nearly linear time algorithm to test for the existence of an H-minor for any fixed H.


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  1. 1.
    Alon, N., Seymour, P., Thomas, R.: A separator theorem for nonplanar graphs. Journal of the American Mathematical Society 3(4), 801–808 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arnborg, S., Proskurowski, A.: Linear time algorithms for np-hard problems restricted to partial k-trees. Discrete Applied Mathematics 23, 11–24 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach. 41, 153–180 (1994)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Dynamic programming algorithms on graphs of bounded tree-width. In: Lepisto, T., Salomaa, A. (eds.) 15th International Colloquium on Automata, Languages and Programming, vol. 317, pp. 105–118. Springer, Heidelberg (1988)Google Scholar
  5. 5.
    Bodlaender, H.L.: Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. Journal of Algorithms 11, 631–643 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Courcelle, B.: The monadic second order logic of graphs. I. recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Demaine, E., Hajiaghayi, M., Nishimura, N., Ragde, P., Thilikos, D.: Approximation algorithms for classes of graphs excluding single-crossing graphs as minors. Journal of Computer and System Sciences 69(2), 166–195 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Demaine, E., Hajiaghayi, M., Thilikos, D.: Exponential speedup of fixed parameter algorithms on k. Algorithmica 41(4), 245–267 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Demaine, E.D., Hajiaghayi, M., Kawarabayashi, K.: Algorithmic graph minor theory: Decomposition, approximation and coloring. In: Proc. 46th Ann. IEEE Symp. Found. Comp. Sci., pp. 637–646 (2005)Google Scholar
  11. 11.
    Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15, 302–318 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gutwenger, C., Mutzel, P.: A linear time implementation of spqr-trees. In: Marks, J. (ed.) Graph Drawing, Colonial Williamsburg, 2000, pp. 77–90. Springer, Heidelberg (2001)Google Scholar
  13. 13.
    Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. Comput. 3, 135–158 (1973)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kawarabayashi, K., Li, Z., Reed, B.: Near linear time algorithms for optimization and recognition for minor closed families (in preparation)Google Scholar
  15. 15.
    Kézdy, A., McGuinness, P.: Sequential and parallel algorithms to find a k5 minor. In: Third Annual Symposium on Discrete Algorithms, pp. 345–356. Springer, Heidelberg (1992)Google Scholar
  16. 16.
    Kuratowski, C.: Sur le problème des courbes gauches en topologie. Fundamenta Mathematica 16, 271–283 (1930)Google Scholar
  17. 17.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36(2), 177–189 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9, 615–627 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Reed, B., Robertson, N., Schrijver, L., Seymour, P.: Finding disjoint trees in planar graphs in linear time. In: Graph Structure Theory, pp. 295–302. AMS (1993)Google Scholar
  20. 20.
    Reed, B., Wood, D.: Fast separation in a graph with an excluded minor. In: EuroConference on Combinatorics, Graph Theory and Applications, pp. 45–50 (2005)Google Scholar
  21. 21.
    Robertson, N., Seymour, P.D.: Graph minors. XIII: the disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Robertson, N., Seymour, P.D.: Graph minors. XVI. excluding a non-planar graph. Journal of Combinatorial Theory, Series B 89, 43–76 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Robertson, N., Seymour, P.D.: Graph minors. XX. wagner’s conjecture. J. Comb. Theory Ser. B 92(2), 325–357 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lagergren, J., Arnborg, S., Seese, D.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12, 308–340 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Wagner, K.: Über eine eigenschaft der ebenen komplexe. Math. Ann. 114, 570–590 (1937)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bruce Reed
    • 1
    • 2
  • Zhentao Li
    • 3
  1. 1.Canada Research Chair in Graph TheoryMcGill UniversityMontrealCanada
  2. 2.Project MascotteINRIACNRS, Sophia-AntipolisFrance
  3. 3.School of Computer ScienceMcGill UniversityMontrealCanada

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