{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,21]],"date-time":"2026-06-21T15:33:10Z","timestamp":1782055990149,"version":"3.54.5"},"reference-count":20,"publisher":"EDP Sciences","issue":"2","license":[{"start":{"date-parts":[[2019,6,5]],"date-time":"2019-06-05T00:00:00Z","timestamp":1559692800000},"content-version":"vor","delay-in-days":65,"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":[[2018,5,30]]},"published-print":{"date-parts":[[2019,4]]},"abstract":"<jats:p>Let <jats:italic>k<\/jats:italic>\u00a0\u2265\u00a01 be an integer and <jats:italic>G<\/jats:italic> be a simple and finite graph with vertex set <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>). A <jats:italic>signed double Roman k-dominating function<\/jats:italic> (SDR<jats:italic>k<\/jats:italic>DF) on a graph <jats:italic>G<\/jats:italic> is a function <jats:italic>f<\/jats:italic>:<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u00a0\u2192\u00a0{\u22121,1,2,3} such that (i) every vertex <jats:italic>v<\/jats:italic> with <jats:italic>f<\/jats:italic>(<jats:italic>v<\/jats:italic>)\u00a0=\u00a0\u22121 is adjacent to at least two vertices assigned a 2 or to at least one vertex <jats:italic>w<\/jats:italic> with <jats:italic>f<\/jats:italic>(<jats:italic>w<\/jats:italic>)\u00a0=\u00a03, (ii) every vertex <jats:italic>v<\/jats:italic> with <jats:italic>f<\/jats:italic>(<jats:italic>v<\/jats:italic>)\u00a0=\u00a01 is adjacent to at least one vertex <jats:italic>w<\/jats:italic> with <jats:italic>f<\/jats:italic>(<jats:italic>w<\/jats:italic>)\u00a0\u2265\u00a02 and (iii) \u2211<jats:sub><jats:italic>u<\/jats:italic>\u2208<jats:italic>N<\/jats:italic>[<jats:italic>v<\/jats:italic>]<\/jats:sub><jats:italic>f<\/jats:italic>(<jats:italic>u<\/jats:italic>)\u00a0\u2265\u00a0<jats:italic>k<\/jats:italic> holds for any vertex <jats:italic>v<\/jats:italic>. The <jats:italic>weight<\/jats:italic> of a SDR<jats:italic>k<\/jats:italic>DF <jats:italic>f<\/jats:italic> is \u2211<jats:sub><jats:italic>u<\/jats:italic>\u2208<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>)<\/jats:sub> <jats:italic>f<\/jats:italic>(<jats:italic>u<\/jats:italic>), and the minimum weight of a SDR<jats:italic>k<\/jats:italic>DF is the <jats:italic>signed double Roman k-domination number<\/jats:italic> <jats:italic>\u03b3<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic><\/jats:sup><jats:sub><jats:italic>sdR<\/jats:italic><\/jats:sub>(<jats:italic>G<\/jats:italic>) of <jats:italic>G<\/jats:italic>. In this paper, we investigate the signed double Roman <jats:italic>k<\/jats:italic>-domination number of trees. In particular, we present lower and upper bounds on <jats:italic>\u03b3<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic><\/jats:sup><jats:sub><jats:italic>sdR<\/jats:italic><\/jats:sub>(<jats:italic>T<\/jats:italic>) for 2\u00a0\u2264\u00a0<jats:italic>k<\/jats:italic>\u00a0\u2264\u00a06 and classify all extremal trees.<\/jats:p>","DOI":"10.1051\/ro\/2018043","type":"journal-article","created":{"date-parts":[[2018,5,31]],"date-time":"2018-05-31T08:23:13Z","timestamp":1527754993000},"page":"627-643","source":"Crossref","is-referenced-by-count":11,"title":["Bounds for signed double Roman <i>k<\/i>-domination in trees"],"prefix":"10.1051","volume":"53","author":[{"given":"Hong","family":"Yang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pu","family":"Wu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sakineh","family":"Nazari-Moghaddam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2298-4744","authenticated-orcid":false,"given":"Seyed Mahmoud","family":"Sheikholeslami","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaosong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zehui","family":"Shao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuan Yan","family":"Tang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"250","published-online":{"date-parts":[[2019,6,5]]},"reference":[{"key":"R1","unstructured":"Abdollahzadeh Ahangar H., Amjadi J., Atapour M., Chellali M. and Sheikholeslami S.M., Double roman trees. 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