On the Outer-Independent Double Roman Domination of Graphs

An outer-independent double Roman dominating function (OIDRDF) of a graph G is h:V(G)→{0,1,2,3} such that i) every vertex v with xmlns:mml="http://www.w3.org/1998/Math/MathML">f(v)=0 adjacent to at least one label 3 or two vertices 2, ii) xmlns:mml="http://www.w3.org/1998/Math/MathML">f(v)=1 greater than 1, and iii) all labeled by 0 are an independent set. The weight OIDRDF the sum its values over vertices. domination number γ oidR ( ) minimum on . It has been shown for any tree T order n ≥ 3, ≤ 5n/4 problem characterizing those trees attaining equality was raised. In this article, we solve give additional bounds number. particular, show that, connected degree in which set three independent, 4n/3.