Total Roman domination and $2$-independence in trees

Abdollahzadeh Ahangar, Hossein; Soroudi, Marzieh; Amjadi, Jafar; Sheikholeslami, Seyed Mahmoud

doi:10.22108/toc.2023.134483.2005

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	Total Roman domination and $2$-independence in trees
Transactions on Combinatorics
مقاله 3، دوره 13، شماره 3، آذر 2024، صفحه 213-223 اصل مقاله (460.04 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2023.134483.2005
نویسندگان
Hossein Abdollahzadeh Ahangar^* ¹؛ Marzieh Soroudi²؛ Jafar Amjadi²؛ Seyed Mahmoud Sheikholeslami²
¹Department of Mathematics Babol Noshirvani University of Technology Shariati Ave., Babol, Iran
²Department of Mathematics Azarbaijan Shahid Madani University Tabriz, Iran
چکیده
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.
کلیدواژه‌ها
total Roman dominating function؛ total Roman domination number؛ $2$-independent set؛ $2$-independence number

مراجع
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Total Roman domination and $2$-independence in trees