From w-Domination in Graphs to Domination Parameters in Lexicographic Product Graphs

Abstract

A wide range of parameters of domination in graphs can be defined and studied through a common approach that was recently introduced in [https://doi.org/10.26493/1855-3974.2318.fb9] under the name of w-domination, where w=(w_0,w_1, dots ,w_l) is a vector of non-negative integers such that w_0ge 1. Given a graph G, a function f: V(G)longrightarrow {0,1,dots ,l} is said to be a w-dominating function if sum _{uin N(v)}f(u)ge w_i for every vertex v with f(v)=i, where N(v) denotes the open neighbourhood of vin V(G). The weight of f is defined to be omega (f)=sum _{vin V(G)} f(v), while the w-domination number of G, denoted by gamma _{w}(G), is defined as the minimum weight among all w-dominating functions on G. A wide range of well-known domination parameters can be defined and studied through this approach. For instance, among others, the vector w=(1,0) corresponds to the case of standard domination, w=(2,1) corresponds to double domination, w=(2,0,0) corresponds to Italian domination, w=(2,0,1) corresponds to quasi-total Italian domination, w=(2,1,1) corresponds to total Italian domination, w=(2,2,2) corresponds to total {2}-domination, while w=(k,k-1,dots ,1,0) corresponds to {k}-domination. In this paper, we show that several domination parameters of lexicographic product graphs Gcirc H are equal to gamma _{w}(G) for some vector win {2}times {0,1,2}^{l} and lin {2,3}. The decision on whether the equality holds for a specific vector w will depend on the value of some domination parameters of H. In particular, we focus on quasi-total Italian domination, total Italian domination, 2-domination, double domination, total {2}-domination, and double total domination of lexicographic product graphs.