Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284220191201A characterization of trees with equal Roman {2}-domination and Roman domination numbers951071385010.22049/cco.2019.26364.1103ENIsmaelGonzalez YeroUniversity of Cadiz0000-0002-1619-1572AbelCabrera MartinezUniversitat Rovira i VirgiliJournal Article20181001Given a graph $G=(V,E)$ and a vertex $v in V$, by $N(v)$ we represent the open neighbourhood of $v$. Let $f:Vrightarrow {0,1,2}$ be a function on $G$. The weight of $f$ is $omega(f)=sum_{vin V}f(v)$ and let $V_i={vin V colon f(v)=i}$, for $i=0,1,2$. The function $f$ is said to be<br /><br />1) a Roman ${2}$-dominating function, if for every vertex $vin V_0$, $sum_{uin N(v)}f(u)geq 2$. The Roman ${2}$-domination number, denoted by $gamma_{{R2}}(G)$, is the minimum weight among all Roman ${2}$-dominating functions on $G$;<br /><br />2) a Roman dominating function, if for every vertex $vin V_0$ there exists $uin N(v)cap V_2$. The Roman domination number, denoted by $gamma_R(G)$, is the minimum weight among all Roman dominating functions on $G$.<br /><br />It is known that for any graph $G$, $gamma_{{R2}}(G)leq gamma_R(G)$. In this paper, we characterize the trees $T$ that satisfy the equality above.http://comb-opt.azaruniv.ac.ir/article_13850_e789cd5a865ff841296b9739ea34aec1.pdf