On the Parameterized Complexity of Grundy Domination and Zero Forcing Problems
Robert Scheffler
TL;DR
This paper studies the parameterized complexity of Grundy domination and zero forcing problems, clarifying a rich landscape of tractability and intractability across several variants. It shows W[1]-completeness for all four Grundy domination variants when parameterized by the dominating sequence size, and extends treewidth-based FPT results to the zero forcing variants and to Grundy domination under a complementary parameter; it also identifies W[1]-hardness for certain related problems. An auxiliary One-Sided Grundy Total Domination result and a detailed treatment of duals under treewidth further map the boundary between feasible and hard cases. Overall, the work integrates new reductions, algorithmic techniques, and open questions to advance understanding of these dual domination and forcing problems.
Abstract
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.