fd}}(G) \le n$, with equality if and only if $G =
{{\overline{K}}}_n$. \[ob1\]

We show later (see Corollary \[c:bound\]) that the result in Observation \[ob1\](b) can be improved as follows: if $G$ is a graph of order $n$, then ${{\rm fd}}(G) \le n-2$, unless $G = {{\overline{K}}}_n$, in which case ${{\rm fd}}(G) = n$, or $G$ contains precisely one edge, in which case ${{\rm fd}}(G) = n-1$.

For example, consider the Petersen graph $G = G_{10}$ shown in Figure \[Peter\] which has domination number $\gamma(G) = 3$. Let $V = V(G)$. The only possible $\gamma(G)$-sets are the open neighborhoods $D = N(v)$, where $v \in V$, but these are not FD-sets since if $u \in V \setminus D$, then either $u \ne v$, in which case $|N(u) \cap D| = 1$, or $u = v$, in which case $|N(u) \cap D| = 3$. Thus, by Observation \[ob1\](a), ${{\rm fd}}(G) > \gamma(G) = 3$. However the closed neighborhood $N[v]$ of any vertex $v \in V$ forms a FD-set in $G$, and so ${{\rm fd}}(G) \le |N[v]| = 4$. Consequently, ${{\rm fd}}(G) = 4$.

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