Total Roman domination and $2$-independence in trees

Document Type : Research Paper


1 Department of Mathematics Babol Noshirvani University of Technology Shariati Ave., Babol, Iran

2 Department of Mathematics Azarbaijan Shahid Madani University Tabriz, Iran


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.


Main Subjects

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Volume 13, Issue 3 - Serial Number 3
September 2024
Pages 213-223
  • Receive Date: 21 July 2022
  • Revise Date: 31 May 2023
  • Accept Date: 08 June 2023
  • Published Online: 01 September 2024