Token Sliding on Graphs of Girth Five
Valentin Bartier, Nicolás Bousquet, Jihad Hanna, Amer E. Mouawad, Sebastian Siebertz · Algorithmica · 2023
Abstract In the Token Sliding problem we are given a graph G and two independent sets $$I_s$$ I s and $$I_t$$ I t in G of size $$k \ge 1$$ k ≥ 1 . The goal is to decide whether there exists a sequence $$\langle I_1, I_2, \ldots , I_\ell \rangle $$ ⟨ I 1 , I 2 , … , I ℓ ⟩ of independent sets such that for all $$j \in \{1,\ldots , \ell - 1\}$$ j ∈ { 1 , … , ℓ - 1 } the set $$I_j$$ I j is an independent set of size k, $$I_1 = I_s$$ I 1 = I s , $$I_\ell = I_t$$ I ℓ = I t and $$I_j \triangle I_{j + 1} = \{u, v\} \in E(G)$$ 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 $$I_s$$ I s into $$I_t$$ 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 -hard on graphs of girth four or less, and the authors posed the question of whether there exists a constant $$p \ge 5$$ 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.