### 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.

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MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2318.fb9

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications