From Italian domination in lexicographic product graphs to w-domination in graphs

In this paper, we show that the Italian domination number of every lexicographic product graph G ○ H can be expressed in terms five different parameters . These defined under following unified approach, which encompasses definition several well-known and introduces new ones. Let N ( v ) denote open neighbourhood ∈ V , let w = 0 1 …, l a vector nonnegative integers such ≥ We say function f : → {0, 1, } is -dominating if )) ∑ u i for vertex with The weight to ω -domination denoted by γ minimum among all functions on Specifically, I where {2} × 2} {2, 3} decision whether equality holds specific values will depend value This paper also provides preliminary results raises challenge conducting detailed study topic.