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.
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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.
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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
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DOI: https://doi.org/10.1007/978-3-319-53925-6_36
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