2$.

\[ob4\] For $m, n \ge 1$, if $G \in \{P_n,K_n,{{\overline{K}}}_n,K_{m,n}\}$, then ${{\rm fd}}(G) = \gamma(G)$. Further for $n \ge 3$, ${{\rm fd}}(C_n) = \gamma(C_n)$ unless $n \equiv 2 \, ({{\rm mod}}\, 3)$ and $n \ge 5$ in which case ${{\rm fd}}(C_n) = \gamma(C_n) + 1$.

We next establish a relationship between the fair domination number and the out-regular number of a graph.

For every graph $G$ of order $n \ge 2$, ${{\rm fd}}(G) + {\xi_{\rm or}}(G) = n$. \[prop1\]

**Proof.** If $G = {{\overline{K}}}_n$, then ${{\rm fd}}(G) = n$ and, by convention, ${\xi_{\rm or}}(G) = 0$. Hence we may assume that $G \ne {{\overline{K}}}_n$, for otherwise the desired result holds. Let $D$ be a ${{\rm fd}}(G)$-set. By Observation \[ob1\](b), ${{\rm fd}}(G) < n$. Let $Q = V \setminus D$. Then, $Q$ is an OR-set in $G$, and so ${\xi_{\rm or}}(G) \ge |Q| = n -
{{\rm fd}}(G)$, or, equivalently, ${{\rm fd}}(G) + {\xi_{\rm or}}(G) \ge n$. Conversely, let $Q$ be an ${\xi_{\rm or}}(G)$-set. By Observation \[ob2\], ${\xi_{\rm or}}(G)
> 0$. By definition, ${\xi_{\rm or}}(G) < n$. Let $D = V \setminus Q$. Then, $D$ is a FD-set, and so ${{\rm fd}}(G) \le |D| = n - {\xi_{\rm or}}(G)$, or, 