Let $G$ be a planar graph and $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ to $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kami\'nski, and Ono to be fixed-parameter tractable. That is, when $k$ is fixed, the problem can be solved in time polynomial in the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a planar graph into an equivalent problem on a (planar) graph with order $O(k)$.

翻译：设$G$为一个平面图，$I_s$和$I_t$为$G$中两个独立集，每个集合的大小均为$k$。我们从$I_s$的每个顶点上放置一个“令牌”开始，通过重复的“令牌跳跃”（即将一个令牌从一个顶点移除并放置到另一个顶点），试图将所有令牌移动到$I_t$。要求每个中间令牌排列仍然构成一个大小为$k$的独立集。给定$G$、$I_s$和$I_t$，我们询问是否存在一组令牌跳跃序列将$I_s$转换为$I_t$。当$k$作为输入的一部分时，该问题已知为PSPACE-完全。然而，Ito、Kamiński和Ono已证明该问题是固定参数可处理的。也就是说，当$k$固定时，该问题可以在关于$G$阶数的多项式时间内求解。本文通过证明该问题具有规模关于$k$线性大小的核心，进一步强化了关于$k$的运行时间上界。更精确地，我们将平面图上的任意输入问题转化为一个阶数为$O(k)$的（平面）图上的等价问题。