%0 Journal Article
%T A characterization relating domination, semitotal domination and total Roman domination in trees
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Cabrera Martinez, Abel
%A Martinez Arias, Alondra
%A Menendez Castillo, Maikel
%D 2021
%\ 12/01/2021
%V 6
%N 2
%P 197-209
%! A characterization relating domination, semitotal domination and total Roman domination in trees
%K total Roman domination
%K semitotal domination
%K domination
%K trees
%R 10.22049/cco.2020.26892.1157
%X A 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.
%U http://comb-opt.azaruniv.ac.ir/article_14113_0b4f6cdb301bc12d4c4eb2b4ad659382.pdf