  The Grundy domination number of a graph $G = (V,E)$ is the length of the
longest sequence of unique vertices $S = (v_1, \ldots, v_k)$ satisfying $N[v_i]
\setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$.
Recently, a generalization of this concept called $k$-Grundy domination was
introduced. In $k$-Grundy domination, a vertex $v$ can be included in $S$ if it
has a neighbor $u$ such that $u$ appears in the closed neighborhood of fewer
than $k$ vertices of $S$. In this paper, we determine the $k$-Grundy domination
number for some families of graphs, find degree-based bounds for the
$k$-$L$-Grundy domination number, and define a relationship between the
$k$-$Z$-Grundy domination number and the $k$-forcing number of a graph.
