Closed formulas for the total Roman domination number of lexicographic product graphs

Abstract

Let G be a graph with no isolated vertex and f: V(G) → {0, 1, 2} a function. Let V_i = {x ∈ V(G) : f(x) = i} for every i ∈ {0, 1, 2}. We say that f is a total Roman dominating function on G if every vertex in V₀ is adjacent to at least one vertex in V₂ and the subgraph induced by V₁ ∪ V₂ has no isolated vertex. The weight of f is ω(f) = ∑_{v ∈ V(G)}f(v). The minimum weight among all total Roman dominating functions on G is the total Roman domination number of G, denoted by γ_tR(G). It is known that the general problem of computing γ_tR(G) is NP-hard. In this paper, we show that if G is a graph with no isolated vertex and H is a nontrivial graph, then the total Roman domination number of the lexicographic product graph G ∘ H is given by γ_tR(G ∘ H) = 2γ_t(G) if γ(H) ≥ 2, and γ_tR(G ∘ H) = ξ(G) if γ(H) = 1, where γ(H) is the domination number of H, γ_t(G) is the total domination number of G and ξ(G) is a domination parameter defined on G.