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

Abstract

For a graph $\Gamma$, let $\gamma (\Gamma ),$ $\gamma _{t}(\Gamma )$, and $\gamma _{tR2}(\Gamma )$ 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 $\Gamma ,$ $\gamma _{t}(\Gamma )\le \gamma _{tR2}(\Gamma )\le 3\gamma (\Gamma ).$ 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 $\gamma _{tR2} (\Gamma )=\gamma _{t}(\Gamma ),$ 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 $\gamma _{tR2}(\Gamma )=3\gamma (\Gamma ),$ and we characterize those graphs that are diameter-2 or trees.