TREES WITH EQUAL ROMAN {2}-DOMINATION NUMBER AND INDEPENDENT ROMAN {2}-DOMINATION NUMBER

摘要

A Roman {2}-dominating function (R{2}DF) on a graph G  =( V ,  E ) is a function f  :  V  → {0, 1, 2} satisfying the condition that every vertex u for which f ( u ) = 0 is adjacent to either at least one vertex v with f ( v ) = 2 or two vertices v _(1), v _(2)with f ( v _(1)) =  f ( v _(2)) = 1. The weight of an R{2}DF f is the value w ( f ) = ∑_(u∈V) f ( u ). The minimum weight of an R{2}DF on a graph G is called the Roman {2}-domination number γ_({ R 2})( G ) of G . An R{2}DF f is called an independent Roman {2}-dominating function (IR{2}DF) if the set of vertices with positive weight under f is independent. The minimum weight of an IR{2}DF on a graph G is called the independent Roman {2}-domination number i _({ R 2})( G ) of G. In this paper, we answer two questions posed by Rahmouni and Chellali.