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. 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 parameterized 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 one of these problems, 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.

翻译：我们研究涉及图支配的两类不同问题族。一方面，我们关注支配序列。在此类序列中，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称为 $\mathsf{Grundy~Domination}$。根据采用闭邻域或开邻域进行支配，该问题存在另外三个变体。我们证明，当以解规模为参数时，所有四个问题变体均为 $\mathsf{W[1]}$-完全问题。另一方面，我们考虑零强迫问题族，它们构成 Grundy 支配问题的参数化对偶。在这些问题中，需要寻找初始着蓝色顶点的最小集合，使得特定的颜色变换规则能够将所有其他顶点着蓝色。Bhyravarapu 等人 [IWOCA 2025] 证明，其中一个称为 $\mathsf{Zero~Forcing~Set}$ 的问题在以树宽或解规模为参数时属于 $\mathsf{FPT}$。我们将他们的树宽结果推广至零强迫的另外三个变体及其对应的 Grundy 支配问题。我们的算法还意味着 $\mathsf{Grundy~Domination}$ 在以未包含在支配序列中的顶点数量为参数时存在 $\mathsf{FPT}$ 算法。