WEAKLY CONNECTED INDEPENDENT 2-DOMINATION IN GRAPHS

Source abstract

Let $G$ be a simple, undirected, nontrivial connected graph. A subset $S \subseteq V(G)$ is called a weakly connected independent 2-dominating set of $G$ if every vertex in $V(G) \backslash S$ is adjacent to at least two vertices of $S$, the vertices of $S$ are pairwise nonadjacent, and the subgraph weakly induced by $S$ is connected. The minimum cardinality of such a set is called the weakly connected independent 2-domination number of $G$, denoted by $i_{2 w}(G)$. In this paper, we characterize weakly connected independent 2-dominating sets in graphs and establish general bounds for the parameter $i_{2 w}(G)$. Exact values of $i_{2 w}(G)$ are determined for > several classes of graphs. We further characterize weakly connected independent 2-dominating sets in the join of graphs and obtain the corresponding weakly connected independent 2-domination number. In addition, realization result is established by showing that, for prescribed integers satisfying suitable conditions, there exists a connected graph whose weakly connected independent 2-domination number attains the specified value.