hand, if $s = r$, then we choose a vertex $v \in X$ and define $D = (V
\setminus X) \cup \{v\}$. Then the set $D$ is a FD-set of $G$, and so ${{\rm fd}}(G) \le n - |X| + 1 \le n - c\log n + 1 = n - c^*\log n$ for some constant $c^*$. [$\Box$]{}

If $G$ is a connected graph on $n \ge 6$ vertices satisfying ${{\rm fd}}(G)
= n - 2$, then $2 \le {{\rm rep}}(G) \le 4$. \[prop5\]

**Proof.** A well-known elementary exercise states that every graph of order at least two has two vertices with the same degree, and so ${{\rm rep}}(G) \ge 2$. Hence it suffices for us to prove that ${{\rm rep}}(G) \le 4$. Assume for the sake of contradiction, that ${{\rm rep}}(G)
\ge 5$. By Ramsey Theory, $r(K_3,K_3) = 6$. Further if $G$ is a graph on five vertices such that neither $G$ nor its complement ${\overline{G}}$ contains a copy of $K_3$, then $G = C_5$. Thus since ${{\rm rep}}(G) \ge 5$ there are either three vertices of the same degree in $G$ that induce an independent set or clique in $G$ or there are five vertices of the same 