An upper bound on triple Roman domination

Hajjari, M.; Abdollahzadeh Ahangar, Hossein; Khoeilar, Rana; Shao, Zehui; Sheikholeslami, S.M.

doi:10.22049/cco.2022.27816.1359

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	An upper bound on triple Roman domination
Communications in Combinatorics and Optimization
مقاله 6، دوره 8، شماره 3، آذر 2023، صفحه 505-511 اصل مقاله (382.43 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2022.27816.1359
نویسندگان
M. Hajjari¹؛ Hossein Abdollahzadeh Ahangar^* ²؛ Rana Khoeilar¹؛ Zehui Shao³؛ S.M. Sheikholeslami¹
¹Azarbaijan Shahid Madani University
²Babol Noshirvani University of Technology
³Guangzhou University
چکیده
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, $\gamma_{[3R]}(G)\leq \frac{3n}{2}$.
کلیدواژه‌ها
Triple Roman dominating function؛ Triple Roman domination number, Trees

مراجع
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An upper bound on triple Roman domination