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∈V f (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.