A simple quadratic kernel for Token Jumping on surfaces
Daniel W. Cranston, Moritz Mühlenthaler, Benjamin Peyrille
TL;DR
The paper addresses Token Jumping, a reconfiguration problem for independent sets, by developing polynomial-time kernelization results for graphs embeddable on fixed surfaces. The authors introduce a partition of the non-target vertices into $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ and bound their sizes using the Heawood bound and Euler's formula, while shrinking $\mathcal{C}_2$ through the construction of small buffers $T_Y$ to obtain an equivalent instance $G'$ of size $O(g^2 + gk + k^2)$. They also derive a sub-quadratic kernel for $K_{2,3}$-free graphs, namely $O(g + \sqrt{g}\,k + k)$, which yields a linear kernel for outerplanar graphs. The results improve on prior kernels that relied on Ramsey-type arguments and demonstrate practical kernel sizes with elementary, embedding-aware techniques. Overall, the work advances kernelization for token reconfiguration on sparse graphs and surfaces, with direct implications for motion-planning-like problems in constrained environments.
Abstract
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.