  A $k$-$\gamma_{c}$-edge critical graph is a graph $G$ with the connected
domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every
$uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be
$k$-$\gamma_{c}$-vertex critical if $\gamma_{c}(G) = k$ and $\gamma_{c}(G - v)
< k$ for all $v \in V(G)$. A maximal $k$-$\gamma_{c}$-vertex critical graph is
a graph which are both $k$-$\gamma_{c}$-edge critical and
$k$-$\gamma_{c}$-vertex critical. Let $\kappa, \delta, \omega$ and $\alpha$ be
respectively connectivity minimum degree, clique number and independence
number. In this paper, we prove that every maximal $3$-$\gamma_{c}$-vertex
critical graph $G$ satisfies $\alpha \leq \delta$ and this bound is best
possible. We prove further that $G$ satisfies $\alpha + \omega \leq n - 1$ and
we also characterize all such graphs achieving the upper bounds. We finally
show that if $G$ satisfies $\kappa < \delta$, then every two vertices of $G$
are joined by hamiltonian path.
