same degree in $G$. Let $H = G[X]$ be the subgraph induced by the set $X$. Clearly, $\chi(H) \le \chi(G)$, and so $\alpha(H) \ge m/\chi(H) \ge m/\chi(G)
\ge n/(2{{\overline{d}}}(G) -2\delta(G) +1)\chi(G)$ by the Caro-West Lemma. Let $Q$ be a maximum independent set in $H$, and so $|Q| = \alpha(H)$. Let $D = V(G) \setminus Q$. Then, $D$ is a FD-set in $G$, and so ${{\rm fd}}(G) \le n - |Q| \le n - \alpha(H)$, and the desired result follows.

\(c) Since $G$ is an $r$-regular graph, we note that ${{\overline{d}}}(G) =
\delta(G)$ and $\chi(G) \le r+1$ and the result follows from Part (b). [$\Box$]{}

Note that, in case that $G$ is a regular graph, the complement of a fair dominating set of $G$ is an induced regular subgraph of $G$. At this point, it is worth mentioning the famous Erdős-Fajtlowicz-Staton problem (see [@AlKrSu]) about the largest induced regular subgraph of a graph $G$. In the case when $G$ is regular, the complement of such a subgraph is a fair dominating set of $G$ .

For $r \ge 1$, if $G$ is an $r$-regular graph on $n$ vertices, then ${{\rm fd}}(G) \le n - c \log n$ for some $c > 