(untitled) (European Journal of Pure and Applied Mathematics, 2025)
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, …, k}, i.e., f : V(G) → P({1, 2, 3, …, k}). If for each vertex v V(G) such that f(v) = , we have _u N_G(v) f(u) = {1, 2, 3, …, k}, then f is called a k-rainbow dominating function (kRDF) of G. A kRDF f : V(G) → P({1,2, …, k}) is said to be a weakly connected k-rainbow dominating function (WCkRDF) if the set S = {v V(G) : f(v) eq } is weakly connected dominating. The weight w(f) of f is defined as ω(f) = _v V(G)f(v) vert. The weakly connected k-rainbow domination number of G, denoted by γ_rk^wc(G) is the minimum weight of WCkRDF. A weakly connected k-rainbow dominating function of G with weight γ^wc_rk(G), i.e., ω(f) = γ^wc_rk(G) is referred to as a γ^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.