Abstract
Let I, J be two given independent sets of a graph G. Imagine that the vertices of an independent set are viewed as tokens (coins). A token is allowed to move (or slide) from one vertex to one of its neighbors. The Sliding Token problem asks whether there exists a sequence of independent sets of G starting from I and ending with J such that each intermediate member of the sequence is obtained from the previous one by moving a token according to the allowed rule. In this paper, we claim that this problem is solvable in polynomial time when the input graph is a block graph—a graph whose blocks are cliques. Our algorithm is developed based on the characterization of a non-trivial structure that, in certain conditions, can be used to indicate a no-instance of the problem. Without such a structure, a sequence of token slidings between any two independent sets of the same cardinality exists.
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
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, Heidelberg (2014). doi:10.1007/978-3-319-08404-6_8
Demaine, E.D., Demaine, M.L., Fox-Epstein, E., Hoang, D.A., Ito, T., Ono, H., Otachi, Y., Uehara, R., Yamada, T.: Linear-time algorithm for sliding tokens on trees. Theor. Comput. Sci. 600, 132–142 (2015)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)
Fox-Epstein, E., Hoang, D.A., Otachi, Y., Uehara, R.: Sliding token on bipartite permutation graphs. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 237–247. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48971-0_21
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), 72–96 (2005)
Hoang, D.A., Uehara, R.: Sliding tokens on a cactus. In: Hong, S.H. (ed.) ISAAC 2016. LIPIcs, vol. 64, pp. 37:1–37:26. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Wadern (2016)
Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci. 412(12), 1054–1065 (2011)
Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theor. Comput. Sci. 439, 9–15 (2012)
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, Heidelberg (2014). doi:10.1007/978-3-319-13524-3_21
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)
Acknowledgement
The first author would like to thank Yota Otachi for his useful discussions. This work is partially supported by MEXT/JSPS Kakenhi Grant Numbers 26330009 and 24106004.
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
Hoang, D.A., Fox-Epstein, E., Uehara, R. (2017). Sliding Tokens on Block Graphs. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_36
Download citation
DOI: https://doi.org/10.1007/978-3-319-53925-6_36
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53924-9
Online ISBN: 978-3-319-53925-6
eBook Packages: Computer ScienceComputer Science (R0)