A simple quadratic kernel for Token Jumping on surfaces

The problem $\text{textsc}TokenJumping$ asks whether, given a graph $G$ and two independent sets of \emph{tokens} $I$ and $J$ of $G$, we can transform $I$ into $J$ by changing the position of a single token in each step and having an independent set of tokens throughout. We show that there is a polynomial-time algorithm that, given an instance of $\text{textsc}TokenJumping$, computes an equivalent instance of size $O(g^2+gk+k^2)$, where $g$ is the genus of the input graph and $k$ is the size of the independent sets.