the following observation.

Let $G$ be a graph of order $n$. Then, ${\xi_{\rm or}}(G) \ge 0$, with equality if and only if $G = {{\overline{K}}}_n$. \[ob2\]

If $S$ is a packing in $G$ (and so, the vertices in $S$ are pairwise at distance at least $3$ apart in $G$), then in order to dominate the vertices in $S$ every dominating set in $G$ must contain at least one vertex in $N[v]$ for each $v \in S$, and so $\gamma(G) \ge |S|$. A *perfect dominating set*, abbreviated PD-set, in $G$ is a dominating set $S$ that is a packing in $G$. Thus if $S$ is a PD-set in $G$, then every vertex not in $S$ is dominated by a unique vertex in $S$, and so $S$ is a 1FD-set, implying that ${{\rm fd}}(G) \le {{\rm fd}_1}(G)
\le \gamma(G)$. Consequently, by Observation \[ob1\], we have the following observation.

If a graph $G$ has a PD-set, then $\gamma(G) = {{\rm fd}_1}(G)  = {{\rm fd}}(G)$. \[ob3\]

Notation
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We denote the *degree* of $v$ in $G$ by $d_G(v)$, or simply by $d(v)$ if the graph $G$ is clear from context. Let $\delta(G)$, $\Delta(G)$ and ${{\overline{d}}}(G)$ denote, respectively, the minimum degree, the maximum degree and 