TY - JOUR ID - 14421 TI - An upper bound on triple Roman domination JO - Communications in Combinatorics and Optimization JA - CCO LA - en SN - 2538-2128 AU - Hajjari, M. AU - Abdollahzadeh Ahangar, Hossein AU - Khoeilar, Rana AU - Shao, Zehui AU - Sheikholeslami, S.M. AD - Azarbaijan Shahid Madani University AD - Babol Noshirvani University of Technology AD - Guangzhou University Y1 - 2023 PY - 2023 VL - 8 IS - 3 SP - 505 EP - 511 KW - Triple Roman dominating function KW - Triple Roman domination number, Trees DO - 10.22049/cco.2022.27816.1359 N2 - For a graph $G=(V,E)$, a triple Roman dominating function (3RD-function) is a function $f:V\to \{0,1,2,3,4\}$ having the property that (i) if $f(v)=0$ then $v$ must have either one neighbor $u$ with $f(u)=4$, or two neighbors $u,w$ with $f(u)+f(w)\ge 5$ or three neighbors $u,w,z$ with $f(u)=f(w)=f(z)=2$, (ii) if $f(v)=1$ then $v$ must have one neighbor $u$ with $f(u)\ge 3$ or two neighbors $u,w$ with $f(u)=f(w)=2$, and (iii) if $f(v)=2$ then $v$ must have one neighbor $u$ with $f(u)\ge 2$. The weight of a 3RDF $f$ is the sum $f(V)=\sum_{v\in V} f(v)$, and the minimum weight of a 3RD-function on $G$ is the triple Roman domination number of $G$, denoted by $\gamma_{[3R]}(G)$. In this paper, we prove that for any connected graph $G$ of order $n$ with minimum degree at least two,  $\gamma_{[3R]}(G)\leq \frac{3n}{2}$. UR - http://comb-opt.azaruniv.ac.ir/article_14421.html L1 - http://comb-opt.azaruniv.ac.ir/article_14421_05e32ae64375bfd597ea59eef02b9e9d.pdf ER -