, Volume 62, Issue 1–2, pp 499–519 | Cite as

The k-in-a-Path Problem for Claw-free Graphs

  • Jiří Fiala
  • Marcin Kamiński
  • Bernard Lidický
  • Daniël PaulusmaEmail author


The k-in-a-Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP-complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw-free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k-Induced Disjoint Paths and the k-in-a-Cycle problem are solvable in polynomial time on claw-free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s i ,t i ) for i=1,…,k and is to test whether G contain k mutually induced paths P i such that P i connects s i and t i for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP-complete, even for the class of line graphs, which form a subclass of the class of claw-free graphs.


Induced path Claw-free graph Polynomial time algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bienstock, D.: On the complexity of testing for odd holes and induced odd paths. Discrete Math. 90, 85–92 (1991). See also Corrigendum, Discrete Math. 102, 109 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chudnovsky, M., Seymour, P.D.: The structure of claw-free graphs. In: Webb, B.S. (ed.) Surveys in Combinatorics, 2005. London Mathematical Society Lecture Notes Series, vol. 327, pp. 153–171. Cambridge University Press, Cambridge (2005) CrossRefGoogle Scholar
  3. 3.
    Chudnovsky, M., Seymour, P.D.: The three-in-a-tree problem. Combinatorica 30, 387–417 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chudnovsky, M., Kawarabayashi, K., Seymour, P.D.: Detecting even holes. J. Graph Theory 48, 85–111 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chudnovsky, M., Robertson, N., Seymour, P.D., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chvátal, V., Sbihi, N.: Recognizing claw-free perfect graphs. J. Comb. Theory, Ser. B 44, 154–176 (1988) CrossRefzbMATHGoogle Scholar
  7. 7.
    Deng, X., Hell, P., Huang, J.: Linear time representation algorithm for proper circular-arc graphs and proper interval graphs. SIAM J. Comput. 25, 390–403 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Derhy, N., Picouleau, C.: Finding induced trees. Discrete Appl. Math. 157, 3552–3557 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Derhy, N., Picouleau, C., Trotignon, N.: The four-in-a-tree problem in triangle-free graphs. Graphs Comb. 25, 489–502 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Faudree, R., Flandrin, E., Ryjáček, Z.: Claw-free graphs—a survey. Discrete Math. 164, 87–147 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fellows, M.R.: The Robertson-Seymour theorems: A survey of applications. In: Richter, R.B. (ed.) Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference. Contemporary Mathematics, vol. 89, pp. 1–18. Am. Math. Soc., Providence (1989) Google Scholar
  12. 12.
    Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, New York (1979) zbMATHGoogle Scholar
  14. 14.
    Golovach, P., Kamiński, M., Paulusma, D., Thilikos, D.M.: Induced packing of odd cycles in a planar graph. In: Dong, Y., Du, D.Z., Ibarra, O.H. (eds.) Proceedings of the 20th International Symposium on Algorithms and Computation, ISAAC 2009. Lecture Notes in Computer Science, vol. 5878, pp. 514–523. Springer, Berlin (2009) Google Scholar
  15. 15.
    Haas, R., Hoffmann, M.: Chordless paths through three vertices. Theor. Comput. Sci. 351, 360–371 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Karp, R.M.: On the complexity of combinatorial problems. Networks 5, 45–68 (1975) zbMATHMathSciNetGoogle Scholar
  17. 17.
    King, A., Reed, B.: Bounding χ in terms of ω and δ for quasi-line graphs. J. Graph Theory 59, 215–228 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kobayashi, Y., Kawarabayashi, K.: The induced disjoint paths problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) Proceedings of the 13th Conference on Integer Programming and Combinatorial Optimization, IPCO 2008. Lecture Notes in Computer Science, vol. 5035, pp. 47–61. Springer, Berlin (2008) Google Scholar
  19. 19.
    Kobayashi, Y., Kawarabayashi, K.: Algorithms for finding an induced cycle in planar graphs and bounded genus graphs. In: Mathieu, C. (ed.) Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pp. 1146–1155. ACM Press, New York (2009) Google Scholar
  20. 20.
    Lévêque, B., Lin, D.Y., Maffray, F., Trotignon, N.: Detecting induced subgraphs. Discrete Appl. Math. 157, 3540–3551 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Liu, W., Trotignon, N.: The k-in-a-tree problem for graphs of girth at least k. Discrete Appl. Math. 158, 1644–1649 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic Press, New York (1969) Google Scholar
  23. 23.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Shrem, S., Stern, M., Golumbic, M.C.: Smallest odd holes in claw-free graphs. In: Paul, C., Habib, M. (eds.) Proceedings of the 35th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2009. Lecture Notes in Computer Science, vol. 5911, pp. 329–340. Springer, Berlin (2009) CrossRefGoogle Scholar
  25. 25.
    van ’t Hof, P., Kamiński, M., Paulusma, D.: Finding induced paths of given parity in claw-free graphs. In: Paul, C., Habib, M. (eds.) Proceedings of the 35th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2009. Lecture Notes in Computer Science, vol. 5911, pp. 341–352. Springer, Berlin (2009) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Jiří Fiala
    • 1
  • Marcin Kamiński
    • 2
  • Bernard Lidický
    • 1
  • Daniël Paulusma
    • 3
    Email author
  1. 1.Faculty of Mathematics and Physics, DIMATIA and Institute for Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic
  2. 2.Computer Science DepartmentUniversité Libre de BruxellesBrusselsBelgium
  3. 3.Department of Computer Science, Science LaboratoriesUniversity of DurhamDurham DH1England

Personalised recommendations