  Let $G$ be a finite simple non-complete connected graph on $\{1, \ldots, n\}$
and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of
free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. Being
motivated by the computation of the depth of the binomial edge ideal of $G$,
the possible sequences $(n, q, f, d)$ of integers for which there is a finite
simple non-complete connected graph $G$ on $\{1, \ldots, n\}$ with $q =
\kappa(G), f = f(G), d = \mathrm{diam}(G)$ satisfying $f + d = n + 2 - q$ will
be determined. Furthermore, finite simple non-complete connected graphs $G$ on
$\{1, \ldots, n\}$ satisfying $f(G) + \mathrm{diam}(G) = n + 2 - \kappa(G)$
will be classified.
