On the secure Italian domination in lexicographic product graphs

In this paper, we study the problem of expressing the secure Italian domination number of lexicographic product graphs [Formula: see text] in terms of a domination parameter of the graph [Formula: see text]. For this purpose, we use the approach of [Formula: see text]-domination introduced in [A. Cabrera Martínez, A. Estrada-Moreno and J. A. Rodríguez-Velázquez, from Italian domination in lexicographic product graphs to [Formula: see text]-domination in graphs, Ars Math. Contemp. 22 (2022) P1.04], where [Formula: see text] is a vector whose components are integers and whose values depend on the value of some domination parameters of the graph [Formula: see text].