Token sliding on graphs of girth five
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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 i ∈{1,…, ℓ} the set I_i is an independent set of size k, I_1 = I_s, I_ℓ = I_t and I_i I_i + 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., 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.
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