, Volume 76, Issue 2, pp 401–425 | Cite as

Planar Disjoint-Paths Completion

  • Isolde Adler
  • Stavros G. Kolliopoulos
  • Dimitrios M. ThilikosEmail author


We introduce Planar Disjoint Paths Completion, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a, not necessarily connected, plane graph Gk pairs of terminals, and a face F of G,  find a minimum-size set of edges, if one exists, to be added inside F so that the embedding remains planar and the pairs become connected by k disjoint paths in the augmented network. Our results are twofold: first, we give an upper bound on the number of necessary additional edges when a solution exists. This bound is a function of k, independent of the size of G. Second, we show that the problem is fixed-parameter tractable, in particular, it can be solved in time \(f(k)\cdot n^{2}\).


Completion problems Disjoint paths Planar graphs 



We wish to thank the anonymous reviewers of an earlier version of this paper for valuable comments and suggestions.


  1. 1.
    Adler, I., Kolliopoulos, S.G., Krause, P.K., Lokshtanov, D., Saurabh, S. and Thilikos, D.M.: Tight bounds for linkages in planar graphs. In: Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011) (2011)Google Scholar
  2. 2.
    Amir, E.: Efficient approximation for triangulation of minimum treewidth. In: Uncertainty in Artificial Intelligence: Proceedings of the Seventeenth Conference (UAI-2001), pp. 7–15. Morgan Kaufmann Publishers, San Francisco, CA (2001)Google Scholar
  3. 3.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theor. Comput. Sci. 109, 49–82 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dawar, A., Grohe, M. and Kreutzer, S.: Locally excluding a minor, in Proc. of the 21st IEEE Symposium on Logic in Computer Science (LICS’07), pp. 270–279. IEEE, New York (2007)Google Scholar
  6. 6.
    Diestel, R.: Graph theory. In: Graduate Texts in Mathematics, vol. 173, 3rd edn. Springer, Berlin (2005)Google Scholar
  7. 7.
    Fomin, F.V., Golovach, P., Thilikos, D.M.: Contraction bidimensionality: the accurate picture. In: 17th Annual European Symposium on Algorithms (ESA ’09), Lecture Notes in Computer Science, pp. 706–717. Springer (2009)Google Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  9. 9.
    Golumbic, M.C., Kaplan, H., Shamir, R.: On the complexity of DNA physical mapping. Adv. Appl. Math. 15, 251–261 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grohe, M., Kawarabayashi, K., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: 43rd ACM Symposium on Theory of Computing (STOC 2011), San Jose, California, June 6–8 (2011)Google Scholar
  11. 11.
    Heggernes, P., Paul, C., Telle, J.A., Villanger, Y.: Interval completion with few edges. In: Proceedings of the 39th Annual ACM Symposium on the Theory of Computing (STOC 2007), pp. 374–381. American Mathematical Society, San Diego, CA (2007)Google Scholar
  12. 12.
    Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput. 28, 1906–1922 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kawarabayashi, K., Wollan, P.: A shorter proof of the graph minor algorithm—the unique linkage theorem. In: Proceedings of the 42nd annual ACM Symposium on Theory of Computing (STOC 2010) (2010)Google Scholar
  14. 14.
    Kramer, M.R., van Leeuven, J.: The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits. Adv. Comput. Res. 2, 129–146 (1984)Google Scholar
  15. 15.
    Reed, B.A.: Finding approximate separators and computing tree width quickly. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp. 221–228. ACM Press (1992)Google Scholar
  16. 16.
    Reed, B.A., Robertson, N., Schrijver, A., Seymour, P.D.: Finding disjoint trees in planar graphs in linear time, in Graph structure theory (Seattle, WA, 1991), vol. 147 of Contemp. Math. Am. Math. Soc., Providence, RI, pp. 295–301 (1993)Google Scholar
  17. 17.
    Robertson, N., Seymour, P.: Graph minors. XXII. Irrelevant vertices in linkage problems, preprint (1992)Google Scholar
  18. 18.
    Robertson, N., Seymour, P.: Graph minors. XXI. Graphs with unique linkages. J. Comb. Theory Ser. B 99, 583–616 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63, 65–110 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Algebr. Discrete Methods 2, 77–79 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Isolde Adler
    • 1
  • Stavros G. Kolliopoulos
    • 2
  • Dimitrios M. Thilikos
    • 3
    • 4
    Email author
  1. 1.Institut für InformatikGoethe-UniversitätFrankfurtGermany
  2. 2.Department of Informatics and TelecommunicationsNational and Kapodistrian University of AthensAthensGreece
  3. 3.Department of MathematicsUniversity of AthensAthensGreece
  4. 4.AlGCo Project TeamCNRS, LIRMMMontpellierFrance

Personalised recommendations