Token Sliding on Graphs of Girth Five

In the Token Sliding problem we are given a graph G and two independent sets \(I_s\) and \(I_t\) in G of size \(k \ge 1\). The goal is to decide whether there exists a sequence \(\langle I_1, I_2, \ldots , I_\ell \rangle \) of independent sets such that for all \(j \in \{1,\ldots , \ell - 1\}\) the set \(I_j\) is an independent set of size k, \(I_1 = I_s\), \(I_\ell = I_t\) and \(I_j \triangle I_{j + 1} = \{u, v\} \in E(G)\). Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the problem asks whether there exists a sequence of independent sets that transforms \(I_s\) into \(I_t\) where at each step we are allowed to slide one token from a vertex to a neighboring vertex. In this paper, we focus on the parameterized complexity of Token Sliding parameterized by k. As shown by Bartier et al. (Algorithmica 83(9):2914–2951, 2021. https://doi.org/10.1007/s00453-021-00848-1), the problem is W[1]-hard on graphs of girth four or less, and the authors posed the question of whether there exists a constant \(p \ge 5\) such that the problem becomes fixed-parameter tractable on graphs of girth at least p. We answer their question positively and prove that the problem is indeed fixed-parameter tractable on graphs of girth five or more, which establishes a full classification of the tractability of Token Sliding parameterized by the number of tokens based on the girth of the input graph.