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

Abstract

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