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 ≥ 1 . The goal is to decide whether there exists a sequence ⟨ I_1, I_2, … , I_ℓ⟩ of independent sets such that for all j ∈{1,… , ℓ - 1} the set I_j is an independent set of size k , I_1 = I_s , I_ℓ = I_t and I_j I_j + 1 = {u, v}∈ 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 ≥ 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.