OUTER-CONNECTED WEAKLY CONNECTED 2 DOMINATION IN GRAPHS
Let $G = (V(G), E(G))$ be a nontrivial connected graph. A subset $S\subseteq V(G)$ is called an outer-connected weakly connected 2-dominating set in $G$ if every vertex $v \in V(G)\setminus S$ is adjacent to at least two vertices in $S$, the subgraph $\langle S \rangle_w$ weakly induced by $S$ is connected, and the induced subgraph $\langle V(G)\setminus S \rangle$ is connected. The minimum cardinality of such a set, denoted by $\gamma_{oc2w}(G)$, is called the outer-connected weakly connected 2-domination number of $G$. In this paper, we introduce and establish the general properties of outer-connected weakly connected 2-dominating set. Exact values of the outer-connected weakly connected 2-domination number of $G$ are obtained for some known families of graphs. The outer-connected weakly connected 2-dominating sets in the join of graphs are characterized and their corresponding numbers are obtained. Furthermore, the parameter is investigated under the vertex corona and edge corona of graphs.