Abstract
Let I be an independent set of a graph G. Imagine that a token is located on every vertex of I. We can now move the tokens of I along the edges of the graph as long as the set of tokens still defines an independent set of G. Given two independent sets I and J, the Token Sliding problem consists in deciding whether there exists a sequence of independent sets which transforms I into J so that every pair of consecutive independent sets of the sequence can be obtained via a single token move. This problem is known to be PSPACE-complete even on planar graphs.
In [9], Demaine et al. asked whether the Token Sliding problem can be decided in polynomial time on interval graphs and more generally on chordal graphs. Yamada and Uehara [20] showed that a polynomial time transformation can be found in proper interval graphs.
In this paper, we answer the first question of Demaine et al. and generalize the result of Yamada and Uehara by showing that we can decide in polynomial time whether an independent set I of an interval graph can be transformed into another independent set J. Moreover, we answer similar questions by showing that: (i) determining if there exists a token sliding transformation between every pair of k-independent sets in an interval graph can be decided in polynomial time; (ii) deciding this latter problem becomes co-NP-hard and even co-W[2]-hard (parameterized by the size of the independent set) on split graphs, a sub-class of chordal graphs.
N. Bousquet—Supported by ANR Projects STINT (ANR-13-BS02-0007) and LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bonamy, M., Bousquet, N.: Recoloring bounded treewidth graphs. Electron. Notes Discret. Math. 44, 257–262 (2013). (LAGOS 2013)
Bonamy, M., Bousquet, N.: Reconfiguring independent sets in cographs. CoRR, abs/1406.1433 (2014)
Bonamy, M., Bousquet, N.: Token sliding on chordal graphs. CoRR, abs/1605.00442 (2016)
Bonamy, M., Bousquet, N., Feghali, C., Johnson, M.: On a conjecture of Mohar concerning Kempe equivalence of regular graphs. CoRR, abs/1510.06964 (2015)
Bonsma, P.: The complexity of rerouting shortest paths. Theoret. Comput. Sci. 510, 1–12 (2013)
Bonsma, P.: Independent set reconfiguration in cographs. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 105–116. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-12340-0_9
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). https://doi.org/10.1007/978-3-319-08404-6_8
Booth, K., Lueker, G.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)
Demaine, E.D., et al.: 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). https://doi.org/10.1007/978-3-319-13075-0_31
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 3rd edn. Springer, Heidelberg (2005)
Feghali, C., Johnson, M., Paulusma, D.: A reconfigurations analogue of Brooks’ theorem and its consequences. CoRR, abs/1501.05800 (2015)
Feghali, C., Johnson, M., Paulusma, D.: Kempe equivalence of colourings of cubic graphs. CoRR, abs/1503.03430 (2015)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York Inc., New York (2006). https://doi.org/10.1007/3-540-29953-X
Gopalan, P., Kolaitis, P.G., Maneva, E., Papadimitriou, C.H.: The connectivity of Boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38(6), 2330–2355 (2009)
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)
Ito, T., Demaine, E., Harvey, N., Papadimitriou, C., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theoret. Comput. Sci. 412(12–14), 1054–1065 (2011)
Kamiński, M., Medvedev, P., Milaniĉ, M.: Complexity of independent set reconfigurability problems. Theoret. Comput. Sci. 439, 9–15 (2012)
Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the parameterized complexity of reconfiguration problems. In: 8th International Symposium on Parameterized and Exact Computation, IPEC 2013, pp. 281–294 (2013)
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, Cambridge (2013)
Yamada, T., Uehara, R.: Shortest reconfiguration of sliding tokens on a caterpillar. In: Kaykobad, M., Petreschi, R. (eds.) WALCOM 2016. LNCS, vol. 9627, pp. 236–248. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30139-6_19
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Bonamy, M., Bousquet, N. (2017). Token Sliding on Chordal Graphs. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-68705-6_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68704-9
Online ISBN: 978-3-319-68705-6
eBook Packages: Computer ScienceComputer Science (R0)