{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T11:15:44Z","timestamp":1770894944394,"version":"3.50.1"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T00:00:00Z","timestamp":1558569600000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T00:00:00Z","timestamp":1558569600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T00:00:00Z","timestamp":1558569600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>We investigate the 2-domination number for grid graphs, that is the size of a smallest set $D$ of vertices of the grid such that each vertex of the grid belongs to $D$ or has at least two neighbours in $D$. We give a closed formula giving the 2-domination number of any $n \\!\\times\\! m$ grid, hereby confirming the results found by Lu and Xu, and Shaheen et al. for $n \\leq 4$ and slightly correct the value of Shaheen et al. for $n = 5$. The proof relies on some dynamic programming algorithms, using transfer matrices in (min,+)-algebra. We also apply the method to solve the Roman domination problem on grid graphs.<\/jats:p><jats:p>Comment: 11 pages, 5 figures, presented at ICGT 2018 The program that led to   the results is included in the Source directory (see Other formats) Accepted   in DMTCS vol 21. Journal version with their template<\/jats:p>","DOI":"10.23638\/dmtcs-21-1-9","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:56:01Z","timestamp":1743699361000},"source":"Crossref","is-referenced-by-count":1,"title":["The 2-domination and Roman domination numbers of grid graphs"],"prefix":"10.23638","volume":"vol. 21 no. 1, ICGT 2018","author":[{"given":"Micha\u00ebl","family":"Rao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandre","family":"Talon","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2019,5,23]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1810.12896v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1810.12896v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:56:01Z","timestamp":1743699361000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/4952"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,23]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/dmtcs-21-1-9","relation":{"has-preprint":[{"id-type":"arxiv","id":"1810.12896v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1810.12896v2","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1810.12896","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1810.12896","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"value":"1365-8050","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,23]]},"article-number":"4952"}}