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.

💡 Research Summary

The paper studies the Token Jumping (TJ) reconfiguration problem: given a graph G and two independent sets I and J of equal size k (the “tokens”), decide whether I can be transformed into J by repeatedly moving a single token to an empty vertex that is non‑adjacent to all other tokens, while maintaining an independent set at every step. This problem has attracted much attention in parameterized complexity, especially when parameterized by the number of tokens k. Prior work established linear or polynomial kernels for planar graphs, bounded‑degree graphs, and K₃,ₜ‑free graphs, but a polynomial kernel parameterized jointly by the genus g of the underlying surface and the token count k was missing.

The authors present a simple, deterministic polynomial‑time kernelization algorithm that, given any instance (G, I, J) with genus g and token count k, produces an equivalent instance whose total number of vertices is bounded by O(g² + gk + k²). Remarkably, the algorithm does not require an explicit embedding of G on a surface of genus g, which is NP‑hard to compute.

The core of the method is a classification of vertices outside the token sets I∪J based on how many neighbors they have in I∪J. For any subset Y⊆I∪J, let C_Y be the set of vertices whose exact neighbor set is Y. The vertices are partitioned into three classes:

• C₁ = {C_Y | |Y| ≤ 1},
• C₂ = {C_Y | |Y| = 2},
• C₃ = {C_Y | |Y| ≥ 3}.

The authors bound the sizes of C₁ and C₃ using classical topological graph theory. By Heawood’s theorem, a graph of genus g can be colored with at most H(g)=⌈(7+√(1+48g))/2⌉ colors. If |C₁| ≥ H(g)·k, then G