We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves, XNL-complete when a maximum length $\ell$ for the sequence is given in binary in the input, and XNLP-complete when $\ell$ is given in unary. The problems were known to be $\mathrm{W}[1]$- and $\mathrm{W}[2]$-hard respectively when $\ell$ is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.

翻译：我们解决了独立集重配置和支配集重配置若干变体的参数化复杂度问题，以令牌数量为参数。研究表明，当移动次数无限制时，这两个问题均为XL-完全；当输入中以二进制形式给出序列最大长度ℓ时，为XNL-完全；当ℓ以一元形式给出时，为XNLP-完全。此前已知当ℓ也作为参数时，这些问题分别属于W[1]-困难和W[2]-困难。我们通过证明这些类别中的成员关系来完善这一结论。此外，我们证明在所考虑的所有变体中，令牌滑动与令牌跳跃在pl-归约下等价。我们引入了令牌跳跃与令牌滑动的一种分区变体，并在四个变体之间建立了能够精确控制令牌数量与重配置序列长度的pl-归约。