In a graph G a sequence v1,v2,…,vm of vertices is Grundy
dominating if for all 2≤i≤m we have N[vi]⊆∪j=1i−1N[vj] and is Grundy total dominating if for all
2≤i≤m we have N(vi)⊆∪j=1i−1N(vj).
The length of the longest Grundy (total) dominating sequence has
been studied by several authors. In this paper we introduce two
similar concepts when the requirement on the neighborhoods is
changed to N(vi)⊆∪j=1i−1N[vj] or
N[vi]⊆∪j=1i−1N(vj). In the former case we
establish a strong connection to the zero forcing number of a graph,
while we determine the complexity of the decision problem in the
latter case. We also study the relationships among the four
concepts, and discuss their computational complexities