{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T17:06:49Z","timestamp":1783962409671,"version":"3.55.0"},"reference-count":43,"publisher":"EDP Sciences","issue":"2","license":[{"start":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T00:00:00Z","timestamp":1746489600000},"content-version":"vor","delay-in-days":66,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2025,3,27]]},"published-print":{"date-parts":[[2025,3]]},"abstract":"<jats:p>Let <jats:italic>G<\/jats:italic> = (<jats:italic>V, E<\/jats:italic>) be a graph of order <jats:italic>n<\/jats:italic>. For <jats:italic>S \u2286 V<\/jats:italic> (<jats:italic>G<\/jats:italic>), the set <jats:italic>N<jats:sub>p<\/jats:sub><\/jats:italic>(<jats:italic>S<\/jats:italic>) is defined as the perfect neighborhood of <jats:italic>S<\/jats:italic> such that all vertices in <jats:italic>V<\/jats:italic> (<jats:italic>G<\/jats:italic>)<jats:italic>\u2216S<\/jats:italic> have exactly one neighbor in <jats:italic>S<\/jats:italic>. The perfect differential of <jats:italic>S<\/jats:italic> is defined to be <jats:italic>\u2202<jats:sub>p<\/jats:sub><\/jats:italic>(<jats:italic>S<\/jats:italic>) = <jats:italic>|N<jats:sub>p<\/jats:sub><\/jats:italic>(<jats:italic>S<\/jats:italic>)| \u2212 <jats:italic>|S|<\/jats:italic> and the perfect differential of a graph is defined as <jats:italic>\u2202<jats:sub>p<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic>) = max{<jats:italic>\u2202<jats:sub>p<\/jats:sub><\/jats:italic>(<jats:italic>S<\/jats:italic>) : <jats:italic>S \u2286 V<\/jats:italic> (<jats:italic>G<\/jats:italic>)}. A perfect Roman dominating function is defined as a Roman dominating function <jats:italic>f<\/jats:italic> satisfying the condition that every vertex <jats:italic>u<\/jats:italic> for which <jats:italic>f<\/jats:italic>(<jats:italic>u<\/jats:italic>) = 0 is adjacent to exactly one vertex <jats:italic>v<\/jats:italic> for which <jats:italic>f<\/jats:italic>(<jats:italic>v<\/jats:italic>) = 2. The perfect Roman domination number, denoted by <jats:italic>\u03b3<jats:sup>p<\/jats:sup><jats:sub>R<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic>), is the minimum weight among all perfect Roman dominating functions on <jats:italic>G<\/jats:italic>, that is <jats:italic>\u03b3<jats:sup>p<\/jats:sup><jats:sub>R<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic>) = min{<jats:italic>w<\/jats:italic>(<jats:italic>f<\/jats:italic>) : <jats:italic>f<\/jats:italic> is a perfect Roman dominating function on <jats:italic>G<\/jats:italic>}. Let <jats:italic>G\u0305<\/jats:italic> be the complement of a Graph <jats:italic>G<\/jats:italic>. The complementary prism <jats:italic>GG\u0305<\/jats:italic> of <jats:italic>G<\/jats:italic> is the graph formed from the disjoint union of <jats:italic>G<\/jats:italic> and <jats:italic>G\u0305<\/jats:italic> by adding the edges of a perfect matching between the corresponding vertices of <jats:italic>G<\/jats:italic> and <jats:italic>G\u0305<\/jats:italic>. This paper is devoted to the computation of perfect differentials of complementary prisms <jats:italic>G<\/jats:italic> <jats:italic>G\u0305<\/jats:italic> and perfect Roman domination numbers of complementary prisms <jats:italic>G<\/jats:italic> <jats:italic>G\u0305<\/jats:italic> by the use of the Gallai-type result proven before. Particular attention is given to the complementary prims of special types of graphs. Furthermore, a sharp lower bound on the perfect differential of the complementary prism <jats:italic>G<\/jats:italic> <jats:italic>G\u0305<\/jats:italic> of a graph <jats:italic>G<\/jats:italic> in terms of the order of <jats:italic>G<\/jats:italic> is presented and the graphs attaining this lower bound are characterized. Finally, the graphs are characterized for which <jats:italic>\u2202<jats:sub>p<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic><jats:italic>G\u0305<\/jats:italic>) and <jats:italic>\u03b3<jats:sup>p<\/jats:sup><jats:sub>R<\/jats:sub><\/jats:italic>(<jats:italic>G<\/jats:italic><jats:italic>G\u0305<\/jats:italic>) are small.<\/jats:p>","DOI":"10.1051\/ro\/2025039","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T08:52:36Z","timestamp":1743151956000},"page":"1247-1256","source":"Crossref","is-referenced-by-count":1,"title":["On the perfect differential and perfect Roman domination in complementary prisms"],"prefix":"10.1051","volume":"59","author":[{"given":"Zeynep Nihan","family":"Berberler","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"250","published-online":{"date-parts":[[2025,5,6]]},"reference":[{"key":"R1","first-page":"218","volume":"68","author":"Alhashim","year":"2017","journal-title":"Aus. 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