even order $\nH = 2n$ defined in Claim I by adding a new vertex $x_{n+1}$ and joining it to every vertex in $X \setminus \{x_1\}$. Let $X_F = X \cup \{x_{n+1}\}$. For the proof, consider an arbitrary FD-set of $F$ and show that $|S| \ge \nF-2$. The proof to show that for an arbitrary FD-set $S$ of $F$ we have $|S| \ge \nF-2$ is similar to the proof presented in Claim I and is therefore omitted. [[($\Box$)]{}]{}

By Claim I and Claim II, there is an infinite family of graphs $G$ of order $n \ge 6$ with $\delta(G) \ge 1$ satisfying ${{\rm fd}}(G) = n - 2$, irrespective of whether $n$ is even or odd. [$\Box$]{}

Note that, by Claims I and II, for each integer $n \ge 6$, there is a connected graph $G$ on $n$ vertices satisfying ${{\rm fd}}(G) = n - 2$. If $n \in \{3,4,5\}$, we can simply take $G = C_n$. Hence for all $n \ge
3$, there exists a connected graph $G$ on $n$ vertices satisfying ${{\rm fd}}(G) = n - 2$.

Let $G$ be a graph of order $n$ with at least two edges. Let $G^*$ be the subgraph obtained from $G$ by 