Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets

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