Weakly Connected $k$-Rainbow Domination in Graphs

Let $G$ be a simple and connected graph, and let $f$ be a function that assigns to each vertex a set of colors chosen from the set $\{1, 2, 3, \dots, k\}$, i.e., $f : V(G) \to \mathcal P(\{1, 2, 3, \dots, k\})$. If for each vertex $v \in V(G)$ such that $f(v) = \varnothing$, we have $\bigcup_{u \in N_G(v)} f(u) = \{1, 2, 3, \dots, k\},$ then $f$ is called a $k$-rainbow dominating function (kRDF) of $G$. A $k$RDF $f : V(G) \to \mathcal P(\{1,2, \dots, k\})$ is said to be a \textit{weakly connected $k$-rainbow dominating function} (WC$k$RDF) if the set $S = \{v \in V(G) : f(v) \neq \varnothing\}$ is weakly connected dominating. The \textit{weight} $w(f)$ of $f$ is defined as $\omega(f) = \sum_{v \in V(G)}\lvert f(v)\rvert$. The \textit{weakly connected $k$-rainbow domination number} of $G$, denoted by $\gamma_{rk}^{wc}(G)$ is the minimum weight of WC$k$RDF. A weakly connected $k$-rainbow dominating function of $G$ with weight $\gamma^{wc}_{rk}(G)$, i.e., $\omega(f) = \gamma^{wc}_{rk}(G)$ is referred to as a $\gamma^{wc}_{rk}$-function of $G$. In this paper, we initiate the study of the weakly connected $k$-rainbow domination parameter. First, we establish fundamental properties and bounds for weakly connected $k$-rainbow domination. Then, we determine the weakly connected $k$-rainbow domination number for various classes of graphs. Furthermore, we characterize the weakly connected $k$-rainbow dominating function under the join of graphs and determine the weakly connected $k$-rainbow domination number for this binary operation.