Lower and upper bounds on independent double Roman domination in trees - M. Kheibari; Hossein Abdollahzadeh Ahangar; R. Khoeilar; S.M. Sheikholeslami (Electronic Journal of Graph Theory and Applications (EJGTA), 2022)
For a graph G = (V, E), a double Roman dominating function (DRDF) f : V → {0, 1, 2, 3} has the property that for every vertex v ∈ V with f(v)=0, either there exists a neighbor u ∈ N(v), with f(u)=3, or at least two neighbors x, y ∈ N(v) having f(x)=f(y)=2, and every vertex with value 1 under f has at least a neighbor with value 2 or 3. The weight of a DRDF is the sum f(V)=∑v ∈ Vf(v). A DRDF f is an independent double Roman dominating function (IDRDF) if the vertices with weight at least two form an independent set. The independent double Roman domination number idR(G) is the minimum weight of an IDRDF on G. In this paper, we show that for every tree T with diameter at least three, i(T)+iR(T)−(s(T))/2 + 1 ≤ idR(T)≤i(T)+iR(T)+s(T)−2, where i(T),iR(T) and s(T) are the independent domination number, the independent Roman domination number and the number of support vertex of T, respectively.