Token Jumping in Minor-Closed Classes

  • Nicolas BousquetEmail author
  • Arnaud Mary
  • Aline Parreau
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)


Given two k-independent sets I and J of a graph G, one can ask if it is possible to transform the one into the other in such a way that, at any step, we replace one vertex of the current independent set by another while keeping the property of being independent. Deciding this problem, known as the Token Jumping (TJ) reconfiguration problem, is PSPACE-complete even on planar graphs. Ito et al. proved in 2014 that the problem is FPT parameterized by k if the input graph is \(K_{3,\ell }\)-free.

We prove that the result of Ito et al. can be extended to any \(K_{\ell ,\ell }\)-free graphs. In other words, if G is a \(K_{\ell ,\ell }\)-free graph, then it is possible to decide in FPT-time if I can be transformed into J. As a by product, the TJ-reconfiguration problem is FPT in many well-known classes of graphs such as any minor-free class.


  1. 1.
    Bonamy, M., Bousquet, N.: Recoloring bounded treewidth graphs. Electron. Notes Discrete Math. (LAGOS 2013) 44, 257–262 (2013)CrossRefGoogle Scholar
  2. 2.
    Bonamy, M., Bousquet, N.: Reconfiguring Independent Sets in Cographs. CoRR, abs/1406.1433 (2014)Google Scholar
  3. 3.
    Bonamy, M., Bousquet, N.: Token sliding on chordal graphs. In: International Workshop on Graph-Theoretic Concepts in Computer Science (WG) (2017, to appear)Google Scholar
  4. 4.
    Bonamy, M., Bousquet, N., Feghali, C., Johnson, M.: On a conjecture of Mohar concerning Kempe equivalence of regular graphs. CoRR, abs/1510.06964 (2015)Google Scholar
  5. 5.
    Bonsma, P.: The complexity of rerouting shortest paths. Theor. Comput. Sci. 510, 1–12 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonsma, P., Kamiński, M., Wrochna, M.: Reconfiguring independent sets in claw-free graphs. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 86–97. Springer, Cham (2014). doi: 10.1007/978-3-319-08404-6_8 CrossRefGoogle Scholar
  7. 7.
    Bousquet, N., Lagoutte, A., Li, Z., Parreau, A., Thomassé, S.: Identifying codes in hereditary classes of graphs and VC-dimension. SIAM J. Discrete Math. 29(4), 2047–2064 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Demaine, E.D., Demaine, M.L., Fox-Epstein, E., Hoang, D.A., Ito, T., Ono, H., Otachi, Y., Uehara, R., Yamada, T.: Polynomial-time algorithm for sliding tokens on trees. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 389–400. Springer, Cham (2014). doi: 10.1007/978-3-319-13075-0_31 Google Scholar
  9. 9.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 3rd edn. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  10. 10.
    Feghali, C., Johnson, M., Paulusma, D.: A reconfigurations analogue of brooks’ theorem and its consequences. CoRR, abs/1501.05800 (2015)Google Scholar
  11. 11.
    Feghali, C., Johnson, M., Paulusma, D.: Kempe equivalence of colourings of cubic graphs. CoRR, abs/1503.03430 (2015)Google Scholar
  12. 12.
    Fredi, Z.: An upper bound on Zarankiewicz’ problem. Comb. Probab. Comput. 5(1), 29–33 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gopalan, P., Kolaitis, P., Maneva, E., Papadimitriou, C.: The connectivity of Boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38, 2330–2355 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hearn, R., Demaine, E.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci. 343(1–2), 72–96 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ito, T., Demaine, E., Harvey, N., Papadimitriou, C., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci. 412(12–14), 1054–1065 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ito, T., Kamiński, M., Ono, H.: Fixed-parameter tractability of token jumping on planar graphs. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 208–219. Springer, Cham (2014). doi: 10.1007/978-3-319-13075-0_17 Google Scholar
  17. 17.
    Ito, T., Kamiński, M., Ono, H., Suzuki, A., Uehara, R., Yamanaka, K.: On the parameterized complexity for token jumping on graphs. In: Gopal, T.V., Agrawal, M., Li, A., Cooper, S.B. (eds.) TAMC 2014. LNCS, vol. 8402, pp. 341–351. Springer, Cham (2014). doi: 10.1007/978-3-319-06089-7_24 CrossRefGoogle Scholar
  18. 18.
    Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theoret. Comput. Sci. 439, 9–15 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kővári, T., Sós, V., Turán, P.: On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57 (1954)zbMATHGoogle Scholar
  20. 20.
    Lokshtanov, D., Mouawad, A.E., Panolan, F., Ramanujan, M.S., Saurabh, S.: Reconfiguration on sparse graphs. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 506–517. Springer, Cham (2015). doi: 10.1007/978-3-319-21840-3_42 CrossRefGoogle Scholar
  21. 21.
    Marx, D.: Efficient approximation schemes for geometric problems? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 448–459. Springer, Heidelberg (2005). doi: 10.1007/11561071_41 CrossRefGoogle Scholar
  22. 22.
    Mouawad, A.E., Nishimura, N., Raman, V., Wrochna, M.: Reconfiguration over tree decompositions. In: Cygan, M., Heggernes, P. (eds.) IPEC 2014. LNCS, vol. 8894, pp. 246–257. Springer, Cham (2014). doi: 10.1007/978-3-319-13524-3_21 Google Scholar
  23. 23.
    Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the parameterized complexity of reconfiguration problems. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 281–294. Springer, Cham (2013). doi: 10.1007/978-3-319-03898-8_24 CrossRefGoogle Scholar
  24. 24.
    Murphy, O.J.: Computing independent sets in graphs with large girth. Discrete Appl. Math. 35(2), 167–170 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    van den Heuvel, J.: The complexity of change. In: Blackburn, S.R., Gerke, S., Wildon, M. (eds.) Surveys in Combinatorics 2013, pp. 127–160. Cambridge University Press (2013)Google Scholar
  26. 26.
    Wrochna, M.: Reconfiguration in bounded bandwidth and treedepth. CoRR, abs/1405.0847 (2014)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Laboratoire G-SCOP, CNRSUniv. Grenoble AlpesGrenobleFrance
  2. 2.Univ Lyon, Université Lyon 1, LBBE CNRS UMR 5558LyonFrance
  3. 3.Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205LyonFrance

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