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Total Domination in Rooted Product Graphs

Secure w-Domination in Graphs

Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
Author to whom correspondence should be addressed.
Symmetry 2020, 12(12), 1948;
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 w01. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of vV(G). We say that a function f:V(G){0,1,,l} is a w-dominating function if f(N(v))=uN(v)f(u)wi for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=vV(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,uV(G) with f(v)=0 and f(u)>0, the function fuv is defined by fuv(v)=1, fuv(u)=f(u)1 and fuv(x)=f(x) for every xV(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 uN(v) such that f(u)>0 and fuv 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
Keywords: secure domination; secure Italian domination; weak roman domination; w-domination secure domination; secure Italian domination; weak roman domination; w-domination
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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.

AMA Style

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

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.

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