Further Progress on the Total Roman {2}-Domination Number of Graphs

For a graphΓ,letγ(Γ), γt (Γ), and γtR2(Γ) denote the domination number, the total domination number, and the total Roman {2}-domination number, respectively. It was shown in Abdollahzadeh Ahangar et al. (Discuss Math Graph Theory, in press) that for each nontrivial connected graph Γ, γt (Γ) ≤ γtR2(Γ) ≤ 3γ(Γ). The problem that arises naturally is to characterize the graphs attaining each bound. For the left inequality, we establish a necessary and sufficient condition for nontrivial connected graphs Γwith γtR2(Γ) = γt (Γ), and we characterize those graphs that are {C3, C6}-free or block. For the right inequality, we present a necessary condition for nontrivial connected graphs Γ with γtR2(Γ) = 3γ(Γ), and we characterize those graphs that are diameter-2 or trees.