Restrained Weakly Connected 2-Domination in the Join of Graphs
Mae P. Militante, Rolito G. Eballe, Rene E. Leonida · Communications in Mathematics and Applications · 2022
\D is dominated by at least two vertices in D and is adjacent to a vertex in V (G)\D, and that the subgraph 〈D〉 w weakly induced by D is connected.The restrained weakly connected 2-domination number of G, denoted by γ r2w (G), is the smallest cardinality of a restrained weakly connected 2-dominating set in G.In this paper, we characterize the RWC2D sets in the join of two graphs G and H, each of which is of order at least 3 and has no isolated vertex, and in the join K 1 ∨ F , where K 1 is the trivial graph and that at least one component of F is of order at least 3.In particular, it is shown that 2 ≤ γ r2w (G ∨ H) ≤ 4 and γ r2w (K 1 ∨ F) = min{1 + γ r (F), γ 2 (F)}, where γ r and γ 2 are the restrained domination and 2-domination parameters, respectively.