A simple quadratic kernel for Token Jumping on surfaces

The problem \textsc{Token Jumping} 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 \textsc{Token Jumping}, 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.