RESTRAINED CONNECTED 2-DOMINATION IN GRAPHS

Let $G = (V(G), E(G))$ be a nontrivial connected graph. A set $S\subseteq V(G)$ is restrained connected 2-dominating set of $G$ if every vertex of $V(G)\setminus S$ is (i) dominated by at least two vertices in $S$ and (ii) adjacent to at least one vertex in $V(G)\setminus S$, and the subgraph $\langle S \rangle$ induced by $S$ is connected. The restrained connected 2-domination number of $G$, denoted by $\gamma_{r2c}(G)$, is the minimum cardinality of restrained connected 2-dominating set of $G$. In this paper, we investigate this parameter and establish Nordhaus-Gaddum type inequalities, extremal results with respect to girth and diameter, and a realization theorem. Moreover, we characterize restrained connected 2-dominating sets in the join and corona of graphs and determine their restrained connected 2-domination numbers.