University of Isfahan Transactions on Combinatorics 2251-8657 13 3 2024 09 01 Total Roman domination and $2$-independence in trees 213 223 27601 10.22108/toc.2023.134483.2005 EN Hossein Abdollahzadeh Ahangar Department of Mathematics Babol Noshirvani University of Technology Shariati Ave., Babol, Iran Marzieh Soroudi Department of Mathematics Azarbaijan Shahid Madani University Tabriz, Iran Jafar Amjadi Department of Mathematics Azarbaijan Shahid Madani University Tabriz, Iran Seyed Mahmoud Sheikholeslami Department of Mathematics Azarbaijan Shahid Madani University Tabriz, Iran Journal Article 2022 07 21 Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. A {\em total Roman dominating function} on a graph $G$ is a function $f:V\rightarrow \{0,1,2\}$ satisfying the following conditions: (i) every vertex $u$ {\color{blue}such that} $f(u)=0$ is adjacent to at least one vertex $v$ {\color{blue}such that} $f(v)=2$ and (ii) the subgraph of $G$ induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function $f$ is the value, $f(V)=\Sigma_{u\in V(G)}f(u)$. The {\em total Roman domination number} $\gamma_{tR}(G)$ of $G$ is the minimum weight of a total Roman dominating function of $G$. A subset $S$ of $V$ is a $2$-independent set of $G$ if every vertex of $S$ has at most one neighbor in $S$. The maximum cardinality of a $2$-independent set of $G$ is the $2$-independence number $\beta_2(G)$. These two parameters are incomparable in general, however, we show that if $T$ is a tree, then $\gamma_{tR}(T)\le \frac{3}{2}\beta_2(T)$ and we characterize all trees attaining the equality. https://toc.ui.ac.ir/article_27601_c165e4cefc9b940f9cbfdfaf670dd9a1.pdf