Towards the Graph Minor Theorems for Directed Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)


Two key results of Robertson and Seymour’s graph minor theory are:
  1. 1.

    a structure theorem stating that all graphs excluding some fixed graph as a minor have a tree decomposition into pieces that are almost embeddable in a fixed surface.

  2. 2.

    the k-disjoint paths problem is tractable when \(k\) is a fixed constant: given a graph \(G\) and \(k\) pairs \((s_1,t_1)\), ..., \((s_k,t_k)\) of vertices of \(G\), decide whether there are \(k\) mutually vertex disjoint paths of \(G\), the \(i\)th path linking \(s_i\) and \(t_i\) for \(i=1,\dots ,k\).


In this talk, we shall try to look at the corresponding problems for digraphs.

Concerning the first point, the grid theorem, originally proved in 1986 by Robertson and Seymour in Graph Minors V, is the basis (even for the whole graph minor project). In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [13, 26]), independently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function \(f\mathrel : {\mathbb {N}} \rightarrow {\mathbb {N}}\) such that every digraph of directed treewidth at least \(f(k)\) contains a directed grid of order \(k\). In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. But for over a decade, this was the most general case proved for the conjecture.

We are finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas conjecture in full generality. As a consequence of our results we are able to improve results in Reed et al. in 1996 [27] to disjoint cycles of length at least \(l\). This would be the first but a significant step toward the structural goals for digraphs (hence towards the first point).

Concerning the second point, in [19] we contribute to the disjoint paths problem using the directed grid theorem. We show that the following can be done in polynomial time:

Suppose that we are given a digraph \(G\) and \(k\) terminal pairs \((s_1, t_1), (s_2, t_2), \dots , (s_k, t_k)\), where \(k\) is a fixed constant. In polynomial time, either
  • we can find \(k\) paths \(P_1, \dots , P_k\) such that \(P_i\) is from \(s_i\) to \(t_i\) for \(i=1, \dots , k\) and every vertex in \(G\) is in at most four of the paths, or

  • we can conclude that \(G\) does not contain disjoint paths \(P_1, \dots , P_k\) such that \(P_i\) is from \(s_i\) to \(t_i\) for \(i=1, \dots , k\).

To the best of our knowledge, this is the first positive result for the general directed disjoint paths problem (and hence for the second point). Note that the directed disjoint paths problem is NP-hard even for \(k=2\). Therefore, this kind of results is the best one can hope for.

We also report some progress on the above two points.


Directed graphs Grid minor The directed disjoint paths problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial \(k\)-trees. Discrete Appl. Math. 23, 11–24 (1989)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decomposition of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chekuri, C., Chuzhoy, J.: Polynomial bounds for the grid-minor theorem. In: Symp. on Theory of Computing (STOC), pp. 60–69 (2014)Google Scholar
  4. 4.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach, in Handbook of Theoretical Computer Science 2, pp. 194–242. Elsevier (1990)Google Scholar
  5. 5.
    Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M.: The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. In: 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 197–206 (2013)Google Scholar
  6. 6.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Bidimensional parameters and local treewidth. SIAM J. Discrete Mathematics 18, 501–511 (2004)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and \(H\)-minor-free graphs. J. ACM 52, 866–893 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Demaine, E.D., Hajiaghayi, M.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Proc. 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 590–601 (2005)Google Scholar
  9. 9.
    Demaine, E.D., Hajiaghayi, M.: Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica 28, 19–36 (2008)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Diestel, R., Gorbunov, K.Y., Jensen, T.R., Thomassen, C.: Highly connected sets and the excluded grid theorem. J. Combin. Theory Ser. B 75, 61–73 (1999)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10, 111–121 (1980)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Grohe, M., Kawarabayashi, K., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: The 43rd ACM Symposium on Theory of Computing (STOC 2011), pp. 479–488Google Scholar
  13. 13.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theory, Ser. B 82(1), 138–154 (2001)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Excluding a grid minor in digraphs (2001). (unpublished manuscript)Google Scholar
  15. 15.
    Kawarabayashi, K., Wollan, P.: A shorter proof of the graph minors algorithm - the unique linkage theorem. In: Proc. 42nd ACM Symposium on Theory of Computing (STOC 2010), pp. 687–694 (2010). A full version of this paper
  16. 16.
    Kawarabayashi, K., Kobayashi, Y.: Linear min-max relation between the treewidth of h-minor-free graphs and its largest grid. In: Dürr, C., Wilke, T. (eds.) STACS, volume 14 of LIPIcs, pp. 278–289. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)Google Scholar
  17. 17.
    Kawarabayashi, K., Kobayashi, Y., Reed, B.: The disjoint paths problem in quadratic time. J. Combin. Theory Ser. B. 102, 424–435 (2012)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kawarabayashi, K., Kreutzer, S.: An excluded grid theorem for digraphs with forbidden minors. In: ACM/SIAM Symposium on Discrete Algorithms (SODA) (2014)Google Scholar
  19. 19.
    Kawarabayashi, K., Kobayashi, Y., Kreutzer, S.: An excluded half-integral grid theorem for digraphs and the directed disjoint paths problem. In: Proc. of the ACM Symposium on Theory of Computing (STOC), pp. 70–78 (2014)Google Scholar
  20. 20.
    Kawarabayashi, K., Kreutzer, S.: The directed excluded grid theorem. In: STOC 2015. arXiv:1411.5681 [cs.DM]
  21. 21.
    Kawarabayashi, K., Reed, B.: A nearly linear time algorithm for the half-integral disjoint paths packing. In: ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 446–454Google Scholar
  22. 22.
    Kleinberg, J.: Decision algorithms for unsplittable flows and the half-disjoint paths problem. In: Proc. 30th ACM Symposium on Theory of Computing (STOC), pp. 530–539 (1998)Google Scholar
  23. 23.
    Leaf, A., Seymour, P.: Treewidth and planar minors (2012)Google Scholar
  24. 24.
    Reed, B.: Finding approximate separators and computing tree width quickly. In: The 24th ACM Symposium on Theory of Computing (STOC 1992)Google Scholar
  25. 25.
    Reed, B.: Tree width and tangles: a new connectivity measure and some applications, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser. 241, pp. 87–162. Cambridge Univ. Press, Cambridge (1997)Google Scholar
  26. 26.
    Reed, B.: Introducing directed tree-width. Electronic Notes in Discrete Mathematics 3, 222–229 (1999)CrossRefGoogle Scholar
  27. 27.
    Reed, B.A., Robertson, N., Seymour, P.D., Thomas, R.: Packing directed circuits. Combinatorica 16(4), 535–554 (1996)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Robertson, N., Seymour, P.D.: Graph minors I - XXIII, 1982–2010. Appearing in Journal of Combinatorial Theory, Series B from 1982 till 2010Google Scholar
  29. 29.
    Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Combin. Theory Ser. B 41, 92–114 (1986)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Robertson, N., Seymour, P.: Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B 63, 65–110 (1995)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Combin. Theory Ser. B 62, 323–348 (1994)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM Jornal on Computing 23(4), 780–788 (1994)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Slivkins, A.: Parameterized tractability of edge-disjoint paths on directed acyclic graphs. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 482–493. Springer, Heidelberg (2003) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.National Institute of InformaticsTokyoJapan
  2. 2.Technical University BerlinBerlinGermany

Personalised recommendations