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

For a graph Gamma, let gamma(Gamma), gamma(t)(Gamma), and gamma(t R2)(Gamma) 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 Gamma, gamma(t)(Gamma) <= gamma(t R2)(Gamma) <= 3 gamma(Gamma). 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 Gamma with gamma(t R2)(Gamma) =gamma t(Gamma), and we characterize those graphs that are {C-3, C-6}free or block. For the right inequality, we present a necessary condition for nontrivial connected graphs Gamma with gamma(t R2)(Gamma) = 3 gamma (Gamma), and we characterize those graphs that are diameter-2 or trees.