# Mixed Roman domination and 2-independence in trees

Document Type : Original paper

Author

Sirjan University of Technology, Sirjan 78137, Iran

Abstract

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A em mixed Roman dominating function (MRDF) of $G$ is a function $f:V\cup E\rightarrow \{0,1,2\}$ satisfying the condition that every element $xin V\cup E$ for which $f(x)=0$ is adjacent‎ ‎or incident to at least one element $y\in V\cup E$ for which $f(y)=2$‎. ‎The weight of an‎ ‎MRDF $f$ is $\sum_{x\in V\cup E} f(x)$‎. ‎The mixed Roman domination number $\gamma^*_R(G)$ of $G$ is‎ ‎the minimum weight among all mixed Roman dominating functions 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 minimum 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 $\frac{4}{3}\beta_2(T)\ge \gamma^*_R(T)$ and we characterize all trees attaining the equality‎.

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