an upper bound on triple roman domination

For a graph g = (v, e), a triple roman dominating function (3rdfunction) is a function f : v → {0, 1, 2, 3, 4} having the property that (i) if f(v) = 0 then v must have either one neighbor u with f(u) = 4, or two neighbors u, w with f(u) + f(w) ≥ 5 or three neighbors u, w, z with f(u) = f(w) = f(z) = 2, (ii) if f(v) = 1 then v must have one neighbor u with f(u) ≥ 3 or two neighbors u, w with f(u) = f(w) = 2, and (iii) if f(v) = 2 then v must have one neighbor u with f(u) ≥ 2. the weight of a 3rdf f is the sum f(v ) = ∑ v∈v f(v), and the minimum weight of a 3rd-function on g is the triple roman domination number of g, denoted by γ[3r] (g). in this paper, we prove that for any connected graph g of order n with minimum degree at least two, γ[3r] (g) ≤ 3n 2 .