Abstract
If G is a graph, then a sequence v1,…,vm of vertices is a closed neighborhood sequence if for all 2≤i≤m we have N[vi]⁄⊆∪j=1i−1N[vj], and it is an open neighborhood sequence if for all 2≤i≤m we have N(vi)⁄⊆∪j=1i−1N(vj). 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(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.
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