Independent double Roman domination in graphs

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 is an element of V with f(v)=0, either there exists a vertex u is an element of N(v), with f(u)=3, or at least two neighbors x,y is an element of 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)=<mml:munder>Sigma v is an element of V</mml:munder>f(v). A DRDF f is called independent if the set of vertices with positive weight under f, is an independent set. The independent double Roman domination number idR(G) is the minimum weight of an independent double Roman dominating function on G. In this paper, we show that for every graph G of order n, ir3(G)-idR(G)<= n/5 and i(G)+iR(G)-idR(G)<= n/4, where ir3(G),iR(G) and i(G) are the independent 3-rainbow domination, independent Roman domination and independent domination numbers, respectively. Moreover, we prove that for any tree G, idR(G)>= ir3(G).