  The Grundy domination number, ${\gamma_{\rm gr}}(G)$, of a graph $G$ is the
maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such
that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$
contains a vertex that does not belong to any closed neighborhood $N[v_j]$,
where $j<i$. It is well known that the Grundy domination number of any graph
$G$ is greater than or equal to the upper domination number $\Gamma(G)$, which
is in turn greater than or equal to the independence number $\alpha(G)$. In
this paper, we initiate the study of the class of graphs $G$ with
$\Gamma(G)={\gamma_{\rm gr}}(G)$ and its subclass consisting of graphs $G$ with
$\alpha(G)={\gamma_{\rm gr}}(G)$. We characterize the latter class of graphs
among all twin-free connected graphs, provide a number of properties of these
graphs, and prove that the hypercubes are members of this class. In addition,
we give several necessary conditions for graphs $G$ with
$\Gamma(G)={\gamma_{\rm gr}}(G)$ and present large families of such graphs.
