{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:25:50Z","timestamp":1760239550694,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,25]],"date-time":"2020-11-25T00:00:00Z","timestamp":1606262400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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,\u2026,wl) is a vector of nonnegative integers such that w0\u22651. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of v\u2208V(G). We say that a function f:V(G)\u27f6{0,1,\u2026,l} is a w-dominating function if f(N(v))=\u2211u\u2208N(v)f(u)\u2265wi for every vertex v with f(v)=i. The weight of f is defined to be \u03c9(f)=\u2211v\u2208V(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,u\u2208V(G) with f(v)=0 and f(u)&gt;0, the function fu\u2192v is defined by fu\u2192v(v)=1, fu\u2192v(u)=f(u)\u22121 and fu\u2192v(x)=f(x) for every x\u2208V(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\u2208N(v) such that f(u)&gt;0 and fu\u2192v is a w-dominating function as well. The secure w-domination number of G, denoted by \u03b3ws(G), is the minimum weight among all secure w-dominating functions. This paper provides fundamental results on \u03b3ws(G) and raises the challenge of conducting a detailed study of the topic.<\/jats:p>","DOI":"10.3390\/sym12121948","type":"journal-article","created":{"date-parts":[[2020,11,25]],"date-time":"2020-11-25T21:55:06Z","timestamp":1606341306000},"page":"1948","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Secure w-Domination in Graphs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2806-4842","authenticated-orcid":false,"given":"Abel Cabrera","family":"Mart\u00ednez","sequence":"first","affiliation":[{"name":"Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9767-2177","authenticated-orcid":false,"given":"Alejandro","family":"Estrada-Moreno","sequence":"additional","affiliation":[{"name":"Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9082-7647","authenticated-orcid":false,"given":"Juan A.","family":"Rodr\u00edguez-Vel\u00e1zquez","sequence":"additional","affiliation":[{"name":"Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cabrera Mart\u00ednez, A., Estrada-Moreno, A., and Rodr\u00edguez-Vel\u00e1zquez, J.A. (2020). From Italian domination in lexicographic product graphs to w-domination in graphs. arXiv.","DOI":"10.1016\/j.dam.2020.03.045"},{"key":"ref_2","first-page":"19","article-title":"Protection of a graph","volume":"67","author":"Cockayne","year":"2005","journal-title":"Util. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1016\/j.ipl.2015.05.006","article-title":"On secure domination in graphs","volume":"115","author":"Chellali","year":"2015","journal-title":"Inform. Process. 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