Mixed Roman domination and 2-independence in trees

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 $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mixed Roman domination number $gamma^*_R(G)$ of $G$ is‎‎the 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.