Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201A characterization relating domination, semitotal domination and total Roman domination in trees1972091411310.22049/cco.2020.26892.1157ENAbelCabrera MartinezUniversitat Rovira i Virgili, Tarragona, Spain0000-0003-2806-4842AlondraMartinez AriasDepartamento de Matemática, Universidad de Oriente, CubaMaikelMenendez CastilloDepartamento de Matemática, Universidad de Oriente, CubaJournal Article20200727A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_{vin V(G)}f(v)$ among all total Roman dominating functions $f$ on $G$. It is known that $gamma_{tR}(G)geq gamma_{t2}(G)+gamma(G)$ for any graph $G$ with neither isolated vertex nor components isomorphic to $K_2$, where $gamma_{t2}(G)$ and $gamma(G)$ represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.http://comb-opt.azaruniv.ac.ir/article_14113_0b4f6cdb301bc12d4c4eb2b4ad659382.pdf