Graphs with small or large Roman {3}-domination number

For an integer k ≥ 1, a Roman {k}-dominating function (R{k}DF) on a graph G = (V, E) is a function f : V → {0, 1, …, k} such that for every vertex v ∈ V with f(v) = 0, ∑u∈N(v) f(u) ≥ k, where N(v) is the set of vertices adjacent to v. The weight of an R{k}DF is the sum of its function values over the whole set of vertices, and the Roman {k}-domination number γ{kR}(G) is the minimum weight of an R{k}DF on G. In this paper, we will be focusing on the case k = 3, where trivially for every connected graphs of order n ≥ 3, 3 ≤ γ{kR}(G) ≤n. We characterize all connected graphs G of order n ≥ 3 such that γ{3R}(G) ∈ {3, n − 1, n}, and we improve the previous lower and upper bounds. Moreover, we show that for every tree T of order n ≥ 3, γ{3R}(T) ≥ γ(T) + 2, where γ(T) is the domination number of T, and we characterize the trees attaining this bound