{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:32:26Z","timestamp":1740133946645,"version":"3.37.3"},"reference-count":18,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62071079"],"award-info":[{"award-number":["62071079"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Found. Comput. Sci."],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:p> Let [Formula: see text] be a graph and [Formula: see text] be an integer representing [Formula: see text] colors. There is a function [Formula: see text] from [Formula: see text] to the power set of [Formula: see text] colors satisfying every vertex [Formula: see text] assigned [Formula: see text] under [Formula: see text] in its neighborhood has all the colors, then [Formula: see text] is called a [Formula: see text]-rainbow dominating function ([Formula: see text]RDF) on [Formula: see text]. The weight of [Formula: see text] is the sum of the number of colors on each vertex all over the graph. The [Formula: see text]-rainbow domination number of [Formula: see text] is the minimum weight of [Formula: see text]RDFs on [Formula: see text], denoted by [Formula: see text]. The aim of this paper is to investigate the [Formula: see text]-rainbow ([Formula: see text]) domination number of the Cartesian product of paths [Formula: see text] and the Cartesian product of paths and cycles [Formula: see text]. For [Formula: see text], we determine the value [Formula: see text] and present [Formula: see text] for [Formula: see text]. For [Formula: see text], we determine the values of [Formula: see text] for [Formula: see text] or [Formula: see text] and [Formula: see text] for [Formula: see text] or [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0129054123500272","type":"journal-article","created":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T09:53:38Z","timestamp":1699955618000},"page":"907-928","source":"Crossref","is-referenced-by-count":1,"title":["Rainbow Domination in Cartesian Product of Paths and Cycles"],"prefix":"10.1142","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8413-2523","authenticated-orcid":false,"given":"Hong","family":"Gao","sequence":"first","affiliation":[{"name":"College of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, Liaoning 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunlei","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, Liaoning 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuqi","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, Liaoning 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanyuan","family":"Guo","sequence":"additional","affiliation":[{"name":"College of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, Liaoning 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xing","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, Liaoning 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Renbang","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, Liaoning 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changqing","family":"Xi","sequence":"additional","affiliation":[{"name":"College of Science, Dalian Maritime University, No. 1 Linghai Road, Dalian, Liaoning 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuansheng","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Dalian University of Technology, No. 2 Linggong Road, Dalian, Liaoning 116024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2023,11,11]]},"reference":[{"key":"S0129054123500272BIB001","doi-asserted-by":"publisher","DOI":"10.11650\/twjm\/1500602498"},{"key":"S0129054123500272BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2009.08.010"},{"key":"S0129054123500272BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2009.03.016"},{"key":"S0129054123500272BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-016-1692-0"},{"key":"S0129054123500272BIB005","first-page":"349","volume":"70","author":"Hao G. 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