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 V 0 is adjacent to at least one vertex in V 2 and the subgraph induced by V 1  ∪  V 2 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 t R ( G ) . It is known that the general problem of computing γ t R ( 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 t R ( G  ∘  H ) = 2 γ t ( G ) if γ ( H ) ≥ 2, and γ t R ( 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 .