Article

# Quadruple Roman Domination in Trees

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Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
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College of Science, East China University of Technology, Nanchang 330013, China
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College of Mathematics and Data Science, Minjiang University, Fuzhou 350108, China
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Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran
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Author to whom correspondence should be addressed.
Academic Editor: Juan Luis García Guirao
Symmetry 2021, 13(8), 1318; https://doi.org/10.3390/sym13081318
Received: 2 July 2021 / Revised: 20 July 2021 / Accepted: 21 July 2021 / Published: 22 July 2021
This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a $\left[k\right]$-Roman dominating function ($\left[k\right]$-RDF) of a simple graph G is a function from the vertex set V of G to the set $\left\{0,1,2,\dots ,k+1\right\}$ if for any vertex $u\in V$ with $f\left(u\right), ${\sum }_{x\in N\left(u\right)\cup \left\{u\right\}}f\left(x\right)\ge |\left\{x\in N\left(u\right):f\left(x\right)\ge 1\right\}|+k$, where $N\left(u\right)$ is the open neighborhood of u. The weight of a $\left[k\right]$-RDF is the value ${\Sigma }_{v\in V}f\left(v\right)$. The minimum weight of a $\left[k\right]$-RDF is called the $\left[k\right]$-Roman domination number ${\gamma }_{\left[kR\right]}\left(G\right)$ of G. In this paper, we establish sharp upper and lower bounds on ${\gamma }_{\left[4R\right]}\left(T\right)$ for nontrivial trees T and characterize extremal trees. View Full-Text
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MDPI and ACS Style

Kou, Z.; Kosari, S.; Hao, G.; Amjadi, J.; Khalili, N. Quadruple Roman Domination in Trees. Symmetry 2021, 13, 1318. https://doi.org/10.3390/sym13081318

AMA Style

Kou Z, Kosari S, Hao G, Amjadi J, Khalili N. Quadruple Roman Domination in Trees. Symmetry. 2021; 13(8):1318. https://doi.org/10.3390/sym13081318

Chicago/Turabian Style

Kou, Zheng, Saeed Kosari, Guoliang Hao, Jafar Amjadi, and Nesa Khalili. 2021. "Quadruple Roman Domination in Trees" Symmetry 13, no. 8: 1318. https://doi.org/10.3390/sym13081318

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