A graph $G$ is said to be $k$-$γ_{c}$-critical if the connected domination number $γ_{c}(G) = k$ and $γ_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Let $δ, κ$ and $α$ be respectively the minimum degree, the connectivity and the independence number. In this paper, we show that a $3$-$γ_{c}$-critical graph $G$ satisfies $α\leq κ+ 2$. Moreover, if $κ\geq 3$, then $α= κ+ p$ if and only if $α= δ+ p$ for all $p \in \{1, 2\}$. We show that the condition $κ+ 1 \leq α\leq κ+ 2$ is best possible to prove that $κ= δ$. By these result, we conclude our paper with an open problem on Hamiltonian connected of $3$-$γ_{c}$-critical graphs.