deleting all isolated vertices in $G$, if any. Then, $\delta(G^*) \ge 1$ and $G^*$ has order $n^* \ge
3$. Applying Theorem \[thm\_n2\] to $G^*$, we have that ${{\rm fd}}(G^*)
\le n^*-2$. Since every ${{\rm fd}}(G^*)$-set can be extended to a FD-set in $G$ by adding to it the set of isolated vertices in $G$, we have that ${{\rm fd}}(G) \le n-2$. Hence as a consequence of Theorem \[thm\_n2\], we have the following result.

If $G$ is a graph of order $n$ and size at least $2$, then ${{\rm fd}}(G)
\le n-2$. \[c:bound\]

We next present some upper bounds on the fair domination number in terms of its order, chromatic number and average, maximum and minimum degrees. For this purpose, we first recall the Caro-Wei Theorem (see [@Ca79; @Wei81]).

[Caro-Wei Theorem.]{} For every graph $G$ of order $n$, $$\alpha(G) \ge \sum_{v \in V(G)} \frac{1}{1 + d_G(v)} \ge
\frac{n}{{{\overline{d}}}(G) + 1}.$$

For our purposes, we also need the following useful lower bound on the repetition number of a graph, established by Caro and West in [@CaWe09].

[Caro-West Lemma.]{} If $G$ is a graph of order $n$, then ${{\rm rep}}(G) \ge n/(2{{\overline{d}}}(G)
-2\delta(G) + 1)$.

Let $G$ be a graph of order $n$. 