Weakly Connected Independent 2-Domination in Graphs

Let \(G\) be a simple, undirected, nontrivial connected graph. A subset \(S ⊆ V(G)\) is called a weakly connected independent 2-dominating set of \(G\) if every vertex in \(V(G) S\) 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 \(i2 w(G)\). In this paper, we characterize weakly connected independent 2-dominating sets in graphs and establish general bounds for the parameter \(i2 w(G)\). Exact values of \(i2 w(G)\) 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.