  In the Token Jumping problem we are given a graph $G = (V,E)$ and two
independent sets $S$ and $T$ of $G$, each of size $k \geq 1$. The goal is to
determine whether there exists a sequence of $k$-sized independent sets in $G$,
$\langle S_0, S_1, \ldots, S_\ell \rangle$, such that for every $i$, $|S_i| =
k$, $S_i$ is an independent set, $S = S_0$, $S_\ell = T$, and $|S_i \Delta
S_{i+1}| = 2$. In other words, if we view each independent set as a collection
of tokens placed on a subset of the vertices of $G$, then the problem asks for
a sequence of independent sets which transforms $S$ to $T$ by individual token
jumps which maintain the independence of the sets. This problem is known to be
PSPACE-complete on very restricted graph classes, e.g., planar bounded degree
graphs and graphs of bounded bandwidth. A closely related problem is the Token
Sliding problem, where instead of allowing a token to jump to any vertex of the
graph we instead require that a token slides along an edge of the graph. Token
Sliding is also known to be PSPACE-complete on the aforementioned graph
classes. We investigate the parameterized complexity of both problems on
several graph classes, focusing on the effect of excluding certain cycles from
the input graph. In particular, we show that both Token Sliding and Token
Jumping are fixed-parameter tractable on $C_4$-free bipartite graphs when
parameterized by $k$. For Token Jumping, we in fact show that the problem
admits a polynomial kernel on $\{C_3,C_4\}$-free graphs. In the case of Token
Sliding, we also show that the problem admits a polynomial kernel on bipartite
graphs of bounded degree. We believe both of these results to be of independent
interest. We complement these positive results by showing that, for any
constant $p \geq 4$, both problems are W[1]-hard on $\{C_4, \dots, C_p\}$-free
graphs and Token Sliding remains W[1]-hard even on bipartite graphs.
