Token sliding on graphs of girth five

In the Token Sliding problem we are given a graph $G$ and two independent sets $Is$ and $It$ in $G$ of size $k \geq 1$. The goal is to decide whether there exists a sequence $\langle I1, I2, \ldots, I\ell \rangle$ of independent sets such that for all $i \in {1,\ldots, \ell}$ the set $Ii$ is an independent set of size $k$, $I1 = Is$, $I\ell = It$ and $Ii \triangle I{i + 1} = {u, v} \in E(G)$. 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 $Is$ into $It$ 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 $p \geq 5$ 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.