Double total domination in the generalized lexicographic product of graphs

Abstract

Let G be a graph of minimum degree at least two. A set D ⊆ V (G) is said to be a double total dominating set of G if |N(v)∩D| ≥ 2 for every  vertex v ∈ V (G). The double total domination number of G, denoted by γ×2,t(G), is the minimum cardinality among all double total  dominating sets of G. In this paper, we study this parameter for any generalized lexicographic product graph G ◦ H. In particular, we show  that γ×2,t(G ◦ H) = γ{2},t(G ◦ H), where γ{2},t(G ◦ H) represents the total {2}-domination number of G ◦ H. Moreover, we obtain tight  bounds and closed formulas on the double total domination number of this specific product of graphs in terms of domination invariants  of the factor graphs. Finally, we characterize the graphs G ◦ H for which γ×2,t(G ◦ H) takes small values.