Doubly connected 2-domination in graphs

For a nontrivial connected graph $G$, a set $C \subseteq V(G)$ is a doubly connected 2-dominating set in $G$ if it is 2-dominating and both $\left\langle C\right\rangle$ and $\left\langle V(G) \backslash C\right\rangle$ are connected. The doubly connected 2-domination number $\gamma_{2cc}(G)$ of $G$, is the smallest cardinality of a doubly connected 2-dominating set in $G$. We prove some properties of the doubly connected 2-dominating set and give bounds to the doubly connected 2-domination number. Also, a connected graph with a given order and doubly connected $2$-domination number is constructed.