We consider two different problem families that deal with domination in graphs. On the one hand, we focus on dominating sequences. In such a sequence, every vertex dominates some vertex of the graph that was not dominated by any earlier vertex in the sequence. The problem of finding the longest dominating sequence is known as $\mathsf{Grundy~Domination}$. Depending on whether the closed or the open neighborhoods are used for domination, there are three other versions of this problem: $\mathsf{Grundy~Total~Domination}$, $\mathsf{L\text{-}Grundy~Domination}$, and $\mathsf{Z\text{-}Grundy~Domination}$. We show that all four problem variants are $\mathsf{W[1]}$-complete when parameterized by the solution size. On the other hand, we consider the family of zero forcing problems which form the parametric duals of the Grundy domination problems. In these problems, one looks for the smallest set of vertices initially colored blue such that certain color change rules are able to color all other vertices blue. Bhyravarapu et al. [IWOCA 2025] showed that the dual of $\mathsf{Z\text{-}Grundy~Domination}$, known as $\mathsf{Zero~Forcing~Set}$, is in $\mathsf{FPT}$ when parameterized by the treewidth or the solution size. We extend their treewidth result to the other three variants of zero forcing and their respective Grundy domination problems. Our algorithm also implies an $\mathsf{FPT}$ algorithm for $\mathsf{Grundy~Domination}$ when parameterized by the number of vertices that are not in the dominating sequence. In contrast, we show that $\mathsf{L\text{-}Grundy~Domination}$ is $\mathsf{W[1]}$-hard for that parameter.

翻译：我们研究了两类与图支配相关的问题族。首先，我们聚焦于支配序列。在这样的序列中，每个顶点支配图中某个未被序列中先前顶点支配的顶点。寻找最长支配序列的问题被称为$\mathsf{Grundy~支配}$。根据支配时使用的是闭邻域还是开邻域，该问题还有三个其他变体：$\mathsf{Grundy~全支配}$、$\mathsf{L\text{-}Grundy~支配}$和$\mathsf{Z\text{-}Grundy~支配}$。我们证明所有这四个问题变体在参数化为解规模时都是$\mathsf{W[1]}$-完全的。其次，我们考虑零强迫问题族，这些问题构成了Grundy支配问题的参数对偶。在这些问题中，目标是寻找最小的初始蓝色顶点集，使得特定的颜色变化规则能够将所有其他顶点染成蓝色。Bhyravarapu等人[IWOCA 2025]证明$\mathsf{Z\text{-}Grundy~支配}$的对偶问题（称为$\mathsf{零强迫集}$）在参数化为树宽或解规模时属于$\mathsf{FPT}$。我们将他们的树宽结果扩展到其他三个零强迫变体及其对应的Grundy支配问题。我们的算法还隐含了一个$\mathsf{Grundy~支配}$在参数化为不在支配序列中的顶点数量时的$\mathsf{FPT}$算法。相反，我们证明$\mathsf{L\text{-}Grundy~支配}$对于该参数是$\mathsf{W[1]}$-难的。