Lower and upper bounds on independent double Roman domination in trees

Abstract

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 i_dR(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)+i_R(T)−(s(T))/2 + 1 ≤ i_dR(T)≤i(T)+i_R(T)+s(T)−2, where i(T),i_R(T) and s(T) are the independent domination number, the independent Roman domination number and the number of support vertex of T, respectively.