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Sliding Tokens on a Cactus

Authors Duc A. Hoang, Ryuhei Uehara

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Duc A. Hoang
Ryuhei Uehara

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Duc A. Hoang and Ryuhei Uehara. Sliding Tokens on a Cactus. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 37:1-37:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ISAAC.2016.37

Abstract

Given two independent sets I and J of a graph G, imagine that a token (coin) is placed on each vertex in I. Then, the Sliding Token problem asks if one could transforms I to J using a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors, such that the resulting set of vertices where tokens are placed still remains independent. In this paper, we describe a polynomial-time algorithm for solving Sliding Token in case the graph G is a cactus. Our algorithm is designed based on two observations. First, all structures that forbid the existence of a sequence of token slidings between I and J, if exist, can be found in polynomial time. A no-instance may be easily deduced using this characterization. Second, without such forbidden structures, a sequence of token slidings between I and J does exist.
Keywords
  • reconfiguration problem
  • token sliding
  • independent set
  • cactus

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