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 d R ( 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 d R ( 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.