If G is a graph, then a sequence v 1 , … , v m of vertices is a closed neighborhood sequence if for all 2 ≤ i ≤ m we have N [ v i ] ⁄ ⊆ ∪ j = 1 i − 1 N [ v j ] , and it is an open neighborhood sequence if for all 2 ≤ i ≤ m we have N ( v i ) ⁄ ⊆ ∪ j = 1 i − 1 N ( v j ) . The length of a longest closed (open) neighborhood sequence is the Grundy (Grundy total) domination number of G . In this paper we introduce two similar concepts in which the requirement on the neighborhoods is changed to N ( v i ) ⁄ ⊆ ∪ j = 1 i − 1 N [ v j ] or N [ v i ] ⁄ ⊆ ∪ j = 1 i − 1 N ( v j ) . 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.