Independence and Connectivity of Connected Domination Critical Graphs

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