An upper bound on triple Roman domination.

For a graph G=(V,E), a triple Roman dominating function (3RD-function) 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∈Vf(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)≤3n2. [ABSTRACT FROM AUTHOR]