Weakly convex 2-domination in graphs

For a nontrivial connected graph $G$, a set $C \subseteq V(G)$ is a weakly convex 2-dominating set in $G$ if $C$ is a 2-dominating and both $C$ is a weakly convex set in $G$. The weakly convex 2-domination number of $G$, denoted by $\gamma_{2cc}(G)$, is the cardinality of a minimum weakly convex 2-dominating set in $G$. We prove some properties of the weakly convex 2-dominating set in a nontrivial connected graph and give bounds to the weakly convex 2-domination number. Also, a connected graph with a given order and weakly convex $2$-domination number is constructed.