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Article

# Secure w-Domination in Graphs

Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
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Symmetry 2020, 12(12), 1948; https://doi.org/10.3390/sym12121948
Received: 31 October 2020 / Revised: 19 November 2020 / Accepted: 23 November 2020 / Published: 25 November 2020
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)
This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where $w=(w0,w1,…,wl)$ is a vector of nonnegative integers such that $w0≥1$. The secure w-domination number is defined as follows. Let G be a graph and $N(v)$ the open neighborhood of $v∈V(G)$. We say that a function $f:V(G)⟶{0,1,…,l}$ is a w-dominating function if $f(N(v))=∑u∈N(v)f(u)≥wi$ for every vertex v with $f(v)=i$. The weight of f is defined to be $ω(f)=∑v∈V(G)f(v)$. Given a w-dominating function f and any pair of adjacent vertices $v,u∈V(G)$ with $f(v)=0$ and $f(u)>0$, the function $fu→v$ is defined by $fu→v(v)=1$, $fu→v(u)=f(u)−1$ and $fu→v(x)=f(x)$ for every $x∈V(G)\{u,v}$. We say that a w-dominating function f is a secure w-dominating function if for every v with $f(v)=0$, there exists $u∈N(v)$ such that $f(u)>0$ and $fu→v$ is a w-dominating function as well. The secure w-domination number of G, denoted by $γws(G)$, is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on $γws(G)$ and raises the challenge of conducting a detailed study of the topic. View Full-Text
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MDPI and ACS Style

Martínez, A.C.; Estrada-Moreno, A.; Rodríguez-Velázquez, J.A. Secure w-Domination in Graphs. Symmetry 2020, 12, 1948. https://doi.org/10.3390/sym12121948

AMA Style

Martínez AC, Estrada-Moreno A, Rodríguez-Velázquez JA. Secure w-Domination in Graphs. Symmetry. 2020; 12(12):1948. https://doi.org/10.3390/sym12121948

Chicago/Turabian Style

Martínez, Abel C., Alejandro Estrada-Moreno, and Juan A. Rodríguez-Velázquez 2020. "Secure w-Domination in Graphs" Symmetry 12, no. 12: 1948. https://doi.org/10.3390/sym12121948

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