{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T13:28:22Z","timestamp":1740144502513,"version":"3.37.3"},"reference-count":15,"publisher":"EDP Sciences","issue":"2","license":[{"start":{"date-parts":[[2019,3,6]],"date-time":"2019-03-06T00:00:00Z","timestamp":1551830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2018,12,2]]},"published-print":{"date-parts":[[2019,4]]},"abstract":"<jats:p>A Roman {2}-dominating function (R{2}DF) on a graph <jats:italic>G<\/jats:italic>\u00a0=(<jats:italic>V<\/jats:italic>,\u00a0<jats:italic>E<\/jats:italic>) is a function <jats:italic>f<\/jats:italic>\u00a0:\u00a0<jats:italic>V<\/jats:italic>\u00a0\u2192\u00a0{0,\u00a01,\u00a02} satisfying the condition that every vertex <jats:italic>u<\/jats:italic> for which <jats:italic>f<\/jats:italic>(<jats:italic>u<\/jats:italic>)\u00a0=\u00a00 is adjacent to either at least one vertex <jats:italic>v<\/jats:italic> with <jats:italic>f<\/jats:italic>(<jats:italic>v<\/jats:italic>)\u00a0=\u00a02 or two vertices <jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub> with <jats:italic>f<\/jats:italic>(<jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>)\u00a0=\u00a0<jats:italic>f<\/jats:italic>(<jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>)\u00a0=\u00a01. The weight of an R{2}DF <jats:italic>f<\/jats:italic> is the value <jats:italic>w<\/jats:italic>(<jats:italic>f<\/jats:italic>)\u00a0=\u00a0\u2211<jats:sub>u\u2208V<\/jats:sub><jats:italic>f<\/jats:italic>(<jats:italic>u<\/jats:italic>). The minimum weight of an R{2}DF on a graph <jats:italic>G<\/jats:italic> is called the Roman {2}-domination number \u03b3<jats:sub>{<jats:italic>R<\/jats:italic>2}<\/jats:sub>(<jats:italic>G<\/jats:italic>) of <jats:italic>G<\/jats:italic>. An R{2}DF <jats:italic>f<\/jats:italic> is called an independent Roman {2}-dominating function (IR{2}DF) if the set of vertices with positive weight under <jats:italic>f<\/jats:italic> is independent. The minimum weight of an IR{2}DF on a graph <jats:italic>G<\/jats:italic> is called the independent Roman {2}-domination number <jats:italic>i<\/jats:italic><jats:sub>{<jats:italic>R<\/jats:italic>2}<\/jats:sub>(<jats:italic>G<\/jats:italic>) of G. In this paper, we answer two questions posed by Rahmouni and Chellali.<\/jats:p>","DOI":"10.1051\/ro\/2018116","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T09:22:28Z","timestamp":1544174548000},"page":"389-400","source":"Crossref","is-referenced-by-count":4,"title":["Trees with equal Roman {2}-domination number and independent Roman {2}-domination number"],"prefix":"10.1051","volume":"53","author":[{"given":"Pu","family":"Wu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zepeng","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0764-4135","authenticated-orcid":false,"given":"Zehui","family":"Shao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2298-4744","authenticated-orcid":false,"given":"Seyed Mahmoud","family":"Sheikholeslami","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2019,3,6]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.dam.2016.03.004","volume":"208","author":"Alvarado","year":"2016","journal-title":"Dis. 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American Mathematical Society, Providence, RI (1967)."},{"key":"R13","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/j.dam.2017.10.028","volume":"236","author":"Rahmouni","year":"2018","journal-title":"Dis. Appl. Math."},{"key":"R14","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.dam.2017.07.026","volume":"233","author":"Shao","year":"2017","journal-title":"Dis. Appl. Math."},{"key":"R15","first-page":"455","volume":"39","author":"Shao","year":"2018","journal-title":"Discuss. Math. 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