Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283120180601Mixed Roman domination and 2-independence in trees79911374710.22049/cco.2018.25964.1062ENNasrinDehgardiSirjan University of Technology, Sirjan 78137, Iran0000-0001-8214-6000Journal Article20170620Let $G=(V, E)$ 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.http://comb-opt.azaruniv.ac.ir/article_13747_2ec084aafd9f82151a96717d372ecd5b.pdf