{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T18:58:07Z","timestamp":1778871487296,"version":"3.51.4"},"reference-count":25,"publisher":"EDP Sciences","issue":"2","license":[{"start":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T00:00:00Z","timestamp":1714694400000},"content-version":"vor","delay-in-days":63,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2024,3,21]]},"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:p>Given a graph <jats:italic>G<\/jats:italic>, we consider the Italian domination number <jats:italic>\u03b3<jats:sub>I<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic>), the 2-rainbow domination number <jats:italic>\u03b3<\/jats:italic><jats:sub><jats:italic>r<\/jats:italic>2<\/jats:sub>(<jats:italic>G<\/jats:italic>) and the Roman domination number <jats:italic>\u03b3<jats:sub>R<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic>). It is known that <jats:italic>\u03b3<jats:sub>I<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic>) \u2264 <jats:italic>\u03b3<\/jats:italic><jats:sub><jats:italic>r<\/jats:italic>2<\/jats:sub>(<jats:italic>G<\/jats:italic>) \u2264 <jats:italic>\u03b3<jats:sub>R<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic>) holds for any graph <jats:italic>G<\/jats:italic>. In this paper, we prove that <jats:italic>\u03b3<jats:sub>I<\/jats:sub><\/jats:italic>(<jats:italic>M<\/jats:italic>(<jats:italic>G<\/jats:italic>)) = <jats:italic>\u03b3<\/jats:italic><jats:sub><jats:italic>r<\/jats:italic>2<\/jats:sub>(<jats:italic>M<\/jats:italic>(<jats:italic>G<\/jats:italic>)) = <jats:italic>\u03b3<jats:sub>R<\/jats:sub><\/jats:italic>(<jats:italic>M<\/jats:italic>(<jats:italic>G<\/jats:italic>)) = <jats:italic>n<\/jats:italic> for the middle graph <jats:italic>M<\/jats:italic>(<jats:italic>G<\/jats:italic>) of a graph <jats:italic>G<\/jats:italic> of order <jats:italic>n<\/jats:italic>, which gives an answer for an open problem posed by Chellali <jats:italic>et al<\/jats:italic>. [<jats:italic>Discrete Appl. Math<\/jats:italic>. <jats:bold>204<\/jats:bold> (2016) 22\u201328]. Moreover, we give a complete characterization of Roman domination stable middle graphs, 2-rainbow domination stable middle graphs and Italian domination stable middle graphs.<\/jats:p>","DOI":"10.1051\/ro\/2024072","type":"journal-article","created":{"date-parts":[[2024,3,25]],"date-time":"2024-03-25T20:01:23Z","timestamp":1711396883000},"page":"2045-2053","source":"Crossref","is-referenced-by-count":1,"title":["Italian, 2-rainbow and Roman domination numbers in middle graphs"],"prefix":"10.1051","volume":"58","author":[{"given":"Kijung","family":"Kim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2024,5,3]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"Alvarado J.D., Dantas S. and Rautenbach D., Averaging 2-rainbow domination and Roman domination. Discrete Appl. 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