Diameter and connectivity of finite simple graphs II

Let G be a finite simple non-complete connected graph on [n] = \1, …, n\ and κ(G) ≥ 1 its vertex connectivity. Let $f(G)$ denote the number of free vertices of G and diam(G) the diameter of G. The final goal of this paper is to determine all sequences of integers $(n,f,d,k)$ with n≥ 8, f≥ 0, d≥ 2 and k≥ 1 for which there exists a finite simple non-complete connected graph on $[n]$ with $f=f(G)$, d=diam(G) and k=κ(G).