Chordless Paths Through Three Vertices

  • Robert Haas
  • Michael Hoffmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3162)


Consider the following problem, that we call “Chordless Path through Three Vertices” or Cp3v, for short: Given a simple undirected graph G=(V,E), a positive integer k, and three distinct vertices s, t, and vV, is there a chordless path from s via v to t in G that consists of at most k vertices? In a chordless path, no two vertices are connected by an edge that is not in the path. Alternatively, one could say that the subgraph induced by the vertex set of the path in G is the path itself. The problem has been raised in the context of service deployment in communication networks. We resolve the parametric complexity of Cp3v by proving it W[1]-complete with respect to its natural parameter k. Our reduction extends to a number of related problems about chordless paths. In particular, deciding on the existence of a single directed chordless (s,t)-path in a digraph is also W[1]-complete with respect to the length of the path.


graph theory induced path parameterized complexity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bazgan, C.: Schémas d’approximation et complexité paramétrée. Rapport du stage (DEA), Université Paris Sud (1995)Google Scholar
  2. 2.
    Berge, C.: Färbung von Graphen deren sämtliche beziehungsweise deren ungerade Kreise starr sind (Zusammenfassung). Wiss. Z.Martin Luther Univ. HalleWittenberg Math. Naturwiss. Reihe, 114–115 (1961)Google Scholar
  3. 3.
    Bienstock, D.: On the complexity of testing for odd holes and induces odd paths. Discrete Math. 90(1), 85–92 (1991)Google Scholar
  4. 4.
    Bienstock, D.: Corrigendum to: On the complexity of testing for odd holes and induces odd paths. Discrete Math. 102(1), 109 (1992)Google Scholar
  5. 5.
    Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: On the parameterized complexity of short computation and factorization. Arch. Math. Logic 36(4–5), 321–337 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cesati, M.: Perfect code is W[1]-complete. Inform. Process. Lett. 81(3), 163–168 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cesati, M.: The Turing way to parameterized complexity. J. Comput. Syst. Sci. 67(4), 654–685 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Inform. Process. Lett. 64(4), 165–171 (1997)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chudnovsky, M., Robertson, N., Seymour, P.D., Thomas, R.: The strong perfect graph theorem (2003) (manuscript)Google Scholar
  10. 10.
    Chudnovsky, M., Seymour, P.D.: Recognizing Berge graphs (2003) (manuscript)Google Scholar
  11. 11.
    CornuéJols, G., Liu, X., Vušković, K.: A polynomial algorithm for recognizing perfect graphs. In: Proc. 44th Annu. IEEE Sympos. Found. Comput. Sci., pp. 20–27 (2003)Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoret. Comput. Sci. 141, 109–131 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Heidelberg (1999)Google Scholar
  14. 14.
    Downey, R.G., Fellows, M.R., Kapron, B., Hallett, M.T., Wareham, H.T.: Parameterized complexity and some problems in logic and linguistics. In: Matiyasevich, Y.V., Nerode, A. (eds.) LFCS 1994. LNCS, vol. 813, pp. 89–101. Springer, Heidelberg (1994)Google Scholar
  15. 15.
    Fellows, M.R.: The Robertson-Seymour theorems: A survey of applications. In: Proc. AMS-IMS-SIAM Joint Summer Research Conf., Providence, RI, pp. 1–18 (1989)Google Scholar
  16. 16.
    Fellows, M.R., Kratochvíl, J., Middendorf, M., Pfeiffer, F.: The complexity of induced minors and related problems. Algorithmica 13, 266–282 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Canad. J. Math. 8, 399–404 (1956)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  20. 20.
    Haas, R.: Service Deployment in Programmable Networks. PhD thesis, ETH Zurich, Switzerland (2003)Google Scholar
  21. 21.
    Karp, R.M.: On the complexity of combinatorial problems. Networks 5, 45–68 (1975)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Lueker, G.S., Rose, D.J., Tarjan, R.E.: Algorithmic aspects of vertex elimination in graphs. SIAM J. Comput. 5, 266–283 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Mcdiarmid, C., Reed, B., Schrijver, A., Shepherd, B.: Non-interfering network flows. In: Proc. 3rd Scand. Workshop Algorithm Theory, pp. 245–257 (1992)Google Scholar
  24. 24.
    Mcdiarmid, C., Reed, B., Schrijver, A., Shepherd, B.: Induced circuits in planar graphs. J. Combin. Theory Ser. B 60, 169–176 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Habilitation thesis, Wilhelm-Schickard Institut für Informatik, Universität Tübingen, Germany (2002)Google Scholar
  26. 26.
    Nikolopoulos, S.D., Palios, L.: Hole and antihole detection in graphs. In: Proc. 15th ACM-SIAM Sympos. Discrete Algorithms, pp. 843–852 (2004)Google Scholar
  27. 27.
    Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63, 65–110 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13, 566–579 (1984)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Robert Haas
    • 1
  • Michael Hoffmann
    • 2
  1. 1.IBM Zurich Research LaboratoryRüschlikon
  2. 2.Institute for Theoretical Computer ScienceETH Zürich 

Personalised recommendations