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A $k$-$\gamma_{c}$-edge critical graph is a graph $G$ with the connecteddomination 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 isa graph which are both $k$-$\gamma_{c}$-edge critical and$k$-$\gamma_{c}$-vertex critical. Let $\kappa, \delta, \omega$ and $\alpha$ berespectively connectivity minimum degree, clique number and independencenumber. In this paper, we prove that every maximal $3$-$\gamma_{c}$-vertexcritical graph $G$ satisfies $\alpha \leq \delta$ and this bound is bestpossible. We prove further that $G$ satisfies $\alpha + \omega \leq n - 1$ andwe also characterize all such graphs achieving the upper bounds. We finallyshow that if $G$ satisfies $\kappa < \delta$, then every two vertices of $G$are joined by hamiltonian path.

Inequalities of Independence Number, Clique Number and Connectivity of Maximal Connected Domination Critical Graphs