{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T09:35:24Z","timestamp":1761212124993,"version":"build-2065373602"},"reference-count":24,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:p>An outer-independent triple Roman dominating function (OI[3]RDF) on a graph [Formula: see text] is function [Formula: see text] having the property that (i) if [Formula: see text] then [Formula: see text] must have either a neighbor assigned 4 or two neighbors one of which is assigned 3 and the other at least 2 or [Formula: see text] has three neighbors all assigned 2; (ii) no two vertices assigned 0 are adjacent; (iii) if [Formula: see text], then [Formula: see text] must have either a neighbor assigned at least 3 or two neighbors assigned 2; (iv) if [Formula: see text], then [Formula: see text] must have one neighbor assigned at least 2. The weight of an OI[3]RDF is the sum of its function value over the whole set of vertices, and the outer-independent triple Roman domination number of [Formula: see text] is the minimum weight of an OI[3]RDF on [Formula: see text] In this paper, we continue the study of outer-independent triple Roman domination number of graphs by first presenting two sharp lower bounds for the outer-independent triple Roman domination number of trees. Then we strengthen the NP-complete result of the outer-independent triple Roman domination problem for bipartite graphs by showing that the problem remains NP-complete for a subclass of bipartite graphs, namely tree convex bipartite graphs, where two special trees are considered.<\/jats:p>","DOI":"10.1142\/s1793830924501234","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T21:14:12Z","timestamp":1731532452000},"source":"Crossref","is-referenced-by-count":0,"title":["More results on the outer-independent triple Roman domination number"],"prefix":"10.1142","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-2796-2818","authenticated-orcid":false,"given":"S.","family":"Babaei","sequence":"first","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9340-4773","authenticated-orcid":false,"given":"J.","family":"Amjadi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5231-6195","authenticated-orcid":false,"given":"M.","family":"Chellali","sequence":"additional","affiliation":[{"name":"LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Blida, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2298-4744","authenticated-orcid":false,"given":"S. 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