{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,23]],"date-time":"2025-03-23T04:21:26Z","timestamp":1742703686398,"version":"3.40.2"},"reference-count":0,"publisher":"Combinatorial Press","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ars Comb."],"published-print":{"date-parts":[[2025,3,30]]},"abstract":"<jats:p>An outer independent double Roman dominating function (OIDRDF) of a graph \\( G \\) is a function \\( f:V(G)\\rightarrow\\{0,1,2,3\\} \\) satisfying the following conditions:\n(i) every vertex \\( v \\) with \\( f(v)=0 \\) is adjacent to a vertex assigned 3 or at least two vertices assigned 2;\n(ii) every vertex \\( v \\) with \\( f(v)=1 \\) has a neighbor assigned 2 or 3;\n(iii) no two vertices assigned 0 are adjacent.\nThe weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number \\( \\gamma_{oidR}(G) \\) is the minimum weight of an OIDRDF on \\( G \\). Ahangar et al. [Appl. Math. Comput. 364 (2020) 124617] established that for every tree \\( T \\) of order \\( n \\geq 4 \\), \\( \\gamma_{oidR}(T)\\leq\\frac{5}{4}n \\) and posed the question of whether this bound holds for all connected graphs. In this paper, we show that for a unicyclic graph \\( G \\) of order \\( n \\), \\( \\gamma_{oidR}(G) \\leq \\frac{5n+2}{4} \\), and for a bicyclic graph, \\( \\gamma_{oidR}(G) \\leq \\frac{5n+4}{4} \\). We further characterize the graphs attaining these bounds, providing a negative answer to the question posed by Ahangar et al.<\/jats:p>","DOI":"10.61091\/ars162-05","type":"journal-article","created":{"date-parts":[[2025,3,22]],"date-time":"2025-03-22T16:54:46Z","timestamp":1742662486000},"page":"51-70","source":"Crossref","is-referenced-by-count":0,"title":["Outer independent double Roman domination in unicyclic and bicyclic graphs"],"prefix":"10.61091","volume":"162","author":[{"name":"Department of Mathematics University of Ilam Ilam, Iran","sequence":"first","affiliation":[]},{"given":"S.","family":"Nazari-Moghaddam","sequence":"first","affiliation":[]},{"given":"M.","family":"Chellali","sequence":"additional","affiliation":[]},{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria","sequence":"additional","affiliation":[]},{"given":"S.M.","family":"Sheikholeslami","sequence":"additional","affiliation":[]},{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran","sequence":"additional","affiliation":[]}],"member":"39747","published-online":{"date-parts":[[2025,3,30]]},"container-title":["Ars Combinatoria"],"original-title":[],"deposited":{"date-parts":[[2025,3,22]],"date-time":"2025-03-22T16:54:52Z","timestamp":1742662492000},"score":1,"resource":{"primary":{"URL":"https:\/\/combinatorialpress.com\/ars-articles\/volume-162\/outer-independent-double-roman-domination-in-unicyclic-and-bicyclic-graphs\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,30]]},"references-count":0,"URL":"https:\/\/doi.org\/10.61091\/ars162-05","relation":{},"ISSN":["0381-7032","2817-5204"],"issn-type":[{"value":"0381-7032","type":"print"},{"value":"2817-5204","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,30]]}}}