Shortest Paths between Shortest Paths and Independent Sets

  • Marcin Kamiński
  • Paul Medvedev
  • Martin Milanič
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)


We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relationships between these models, observing that shortest path reconfiguration is a special case of independent set reconfiguration in perfect graphs, under any of the three models. Finally, we give polynomial results for restricted classes of graphs (even-hole-free and P 4-free graphs).


Short Path Feasible Solution Polynomial Time Line Graph Input Graph 
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  1. 1.
    Bonsma, P.S., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theor. Comput. Sci. 410(50), 5215–5226 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bonsma, P.S., Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between graph colourings: Computational complexity and possible distances. Electronic Notes in Discrete Mathematics 29, 463–469 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cereceda, L., van den Heuvel, J., Johnson, M.: Connectedness of the graph of vertex-colourings. Discrete Mathematics 308(5-6), 913–919 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cereceda, L., van den Heuvel, J., Johnson, M.: Mixing 3-colourings in bipartite graphs. European Journal of Combinatorics 30, 1593–1606 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. of Math. 164, 51–229 (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Conforti, M., Cornuéjols, G., Kapoor, A., Vušković, K.: Even-hole-free graphs part II: Recognition algorithm. J. Graph Theory 40, 238–266 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Corneil, D.G., Lerchs, H., Stewart Burlingham, L.: Complement reducible graphs. Discrete Applied Mathematics 3(3), 163–174 (1981)CrossRefzbMATHGoogle Scholar
  8. 8.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14(4), 926–934 (1985)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.H.: The connectivity of Boolean satisfiability: Computational and structural dichotomies. SIAM J. Comput. 38(6), 2330–2355 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hearn, R.A., Demaine, E.D.: 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)CrossRefzbMATHGoogle Scholar
  11. 11.
    Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 28–39. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Ito, T., Kamiński, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 375–386. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Kaminski, M., Medvedev, P., Milanic, M.: Shortest paths between shortest paths and independent sets. CoRR, abs/1008.4563 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Marcin Kamiński
    • 1
  • Paul Medvedev
    • 2
  • Martin Milanič
    • 3
  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  2. 2.Department of Computer ScienceUniversity of TorontoTorontoCanada
  3. 3.FAMNIT and PINTUniversity of PrimorskaKoperSlovenia

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