  A graph $G=(V,E)$ is $\gamma$-excellent if $V$ is a union of all
$\gamma$-sets of $G$, where $\gamma$ stands for the domination number. Let
$\mathcal{I}$ be a set of all mutually nonisomorphic graphs and $\emptyset
\not= \mathcal{H} \subsetneq \mathcal{I}$. In this paper we initiate the study
of the $\mathcal{H}$-$\gamma$-excellent graphs, which we define as follows. A
graph $G$ is $\mathcal{H}$-$\gamma$-excellent if the following hold: (i) for
every $H \in \mathcal{H}$ and for each $x \in V(G)$ there exists an induced
subgraph $H_x$ of $G$ such that $H$ and $H_x$ are isomorphic, $x \in V(H_x)$
and $V(H_x)$ is a subset of some $\gamma$-set of $G$, and (ii) the vertex set
of every induced subgraph $H$ of $G$, which is isomorphic to some element of
$\mathcal{H}$, is a subset of some $\gamma$-set of $G$. For each of some well
known graphs, including cycles, trees and some cartesian products of two
graphs, we describe its largest set $\mathcal{H} \subsetneq \mathcal{I}$ for
which the graph is $\mathcal{H}$-$\gamma$-excellent. Results on
$\gamma$-excellent regular graphs and a generalized lexicographic product of
graphs are presented. Several open problems and questions are posed.
