Author, as appears in the article.: Martínez AC; Estrada-Moreno A; Rodríguez-Velázquez JA

Department: Enginyeria Informàtica i Matemàtiques

e-ISSN: 2073-8994

URV's Author/s: CABRERA MARTÍNEZ, ABEL / Estrada Moreno, Alejandro / Rodríguez Velázquez, Juan Alberto

Keywords: Weak roman domination W-domination Secure italian domination Secure domination

Abstract: © 2020 by the authors. Licensee MDPI, Basel, Switzerland. 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 γw(G),s is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on γw(G)s and raises the challenge of conducting a detailed study of the topic.

Thematic Areas: Visual arts and performing arts Physics and astronomy (miscellaneous) Multidisciplinary sciences Modeling and simulation Mathematics, interdisciplinary applications Mathematics (miscellaneous) Mathematics (all) Matemática / probabilidade e estatística General mathematics Engineering (miscellaneous) Computer science (miscellaneous) Ciência da computação Chemistry (miscellaneous) Arts and humanities (miscellaneous) Architecture Applied mathematics

licence for use: https://creativecommons.org/licenses/by/3.0/es/

Author's mail: alejandro.estrada@urv.cat juanalberto.rodriguez@urv.cat

Author identifier: 0000-0001-9767-2177 0000-0002-9082-7647

Record's date: 2023-08-05

Papper version: info:eu-repo/semantics/publishedVersion

Link to the original source: https://www.mdpi.com/2073-8994/12/12/1948

Licence document URL: http://repositori.urv.cat/ca/proteccio-de-dades/

Papper original source: Symmetry-Basel. 12 (12): 1-11

APA: Martínez AC; Estrada-Moreno A; Rodríguez-Velázquez JA (2020). Secure w-domination in graphs. Symmetry-Basel, 12(12), 1-11. DOI: 10.3390/sym12121948

Article's DOI: 10.3390/sym12121948

Entity: Universitat Rovira i Virgili

Journal publication year: 2020

Publication Type: Journal Publications