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

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, … ,w_l) is a vector of non-negative integers such that w_0≥ 1 . Given a graph G , a function f: V(G)⟶{0,1,… ,l} is said to be a w -dominating function if ∑ _u∈ N(v)f(u)≥ w_i for every vertex v with f(v)=i , where N ( v ) denotes the open neighbourhood of v∈ V(G) . The weight of f is defined to be ω (f)=∑ _v∈ V(G) f(v) , while the w -domination number of G , denoted by γ _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,… ,1,0) corresponds to {k} -domination. In this paper, we show that several domination parameters of lexicographic product graphs G∘ H are equal to γ _w(G) for some vector w∈{2}×{0,1,2}^l and l∈{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.