Independence Number and Connectivity of Maximal Connected Domination Vertex Critical Graphs

A $k$-CEC graph is a graph $G$ which has connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. A $k$-CVC graph $G$ is a $2$-connected graph with  $\gamma_{c}(G) = k$ and $\gamma_{c}(G - v) < k$ for any $v \in V(G)$. A graph is said to be maximal $k$-CVC if it is both $k$-CEC and $k$-CVC. Let $\delta$, $\kappa$, and $\alpha$ be the minimum degree, connectivity, and independence number of $G$, respectively. In this work, we prove that for a maximal $3$-CVC graph, if $\alpha = \kappa$, then $\kappa = \delta$. We additionally consider the class of maximal $3$-CVC graphs with $\alpha < \kappa$ and $\kappa < \delta$, and prove that every $3$-connected maximal $3$-CVC graph when $\kappa < \delta$ is Hamiltonian connected.