{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T11:51:40Z","timestamp":1771329100991,"version":"3.50.1"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T00:00:00Z","timestamp":1672099200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T00:00:00Z","timestamp":1672099200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s00373-022-02600-0","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T16:02:41Z","timestamp":1672156961000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Independent Domination in Direct Products"],"prefix":"10.1007","volume":"39","author":[{"given":"Kirsti","family":"Kuenzel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5482-756X","authenticated-orcid":false,"given":"Douglas F.","family":"Rall","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,27]]},"reference":[{"issue":"2","key":"2600_CR1","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/0012-365X(78)90105-X","volume":"23","author":"Robert Allan","year":"1978","unstructured":"Allan, Robert, Laskar, Renu: On domination and independent domination numbers of a graph. Discrete Math. 23(2), 73\u201376 (1978)","journal-title":"Discrete Math."},{"issue":"1","key":"2600_CR2","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1002\/jgt.20194","volume":"54","author":"Noga Alon","year":"2007","unstructured":"Alon, Noga, Lubetzky, Eyal: Independent sets in tensor graph powers. J. Graph Theory 54(1), 73\u201387 (2007)","journal-title":"J. Graph Theory"},{"key":"2600_CR3","doi-asserted-by":"crossref","unstructured":"Bre\u0161ar, Bo\u0161tjan.: Vizing-like conjecture for the upper domination of Cartesian products of graphs-the proof. Electron. J. Combin. 12, Note 12, 6 pp. (2005)","DOI":"10.37236\/1979"},{"issue":"13","key":"2600_CR4","doi-asserted-by":"publisher","first-page":"1636","DOI":"10.1016\/j.disc.2006.09.013","volume":"307","author":"Bo\u0161tjan Bre\u0161ar","year":"2007","unstructured":"Bre\u0161ar, Bo\u0161tjan, Klav\u017ear, Sandi, Rall, Douglas F.: Dominating direct products of graphs. Discrete Math. 307(13), 1636\u20131642 (2007)","journal-title":"Discrete Math."},{"issue":"2","key":"2600_CR5","doi-asserted-by":"publisher","first-page":"290","DOI":"10.1137\/S0895480194276909","volume":"9","author":"Jason I Brown","year":"1996","unstructured":"Brown, Jason I., Nowakowski, Richard J., Rall, Douglas: The ultimate categorical independence ratio of a graph. SIAM J. Discrete Math. 9(2), 290\u2013300 (1996)","journal-title":"SIAM J. Discrete Math."},{"key":"2600_CR6","unstructured":"Burcroff, Amanda.: Personal Communication. (2022)"},{"key":"2600_CR7","first-page":"205","volume":"99","author":"Louis M Friedler","year":"2011","unstructured":"Friedler, Louis M.: The independence number of the Cartesian product of graphs. Ars Combin. 99, 205\u2013216 (2011)","journal-title":"Ars Combin."},{"issue":"7","key":"2600_CR8","doi-asserted-by":"publisher","first-page":"839","DOI":"10.1016\/j.disc.2012.11.031","volume":"313","author":"Wayne Goddard","year":"2013","unstructured":"Goddard, Wayne, Henning, Michael A.: Independent domination in graphs: a survey and recent results. Discrete Math. 313(7), 839\u2013854 (2013)","journal-title":"Discrete Math."},{"key":"2600_CR9","first-page":"149","volume":"43","author":"Johann Hagauer","year":"1996","unstructured":"Hagauer, Johann, Klav\u017ear, Sandi: On independence numbers of the Cartesian product of graphs. Ars Combin. 43, 149\u2013157 (1996)","journal-title":"Ars Combin."},{"key":"2600_CR10","doi-asserted-by":"crossref","unstructured":"Hell, Pavol, Yu, Xingxing, Zhou, Hui Shan: Independence ratios of graph powers. Graph theory and applications (Hakone, 1990). Discrete Math. 127, no. 1-3, 213\u2013220 (1994)","DOI":"10.1016\/0012-365X(92)00480-F"},{"key":"2600_CR11","first-page":"53","volume":"50","author":"Pranava K Jha","year":"1998","unstructured":"Jha, Pranava K., Klav\u017ear, Sandi: Independence in direct-product graphs. Ars Combin. 50, 53\u201363 (1998)","journal-title":"Ars Combin."},{"issue":"4","key":"2600_CR12","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/0893-9659(94)90018-3","volume":"7","author":"PK Jha","year":"1994","unstructured":"Jha, P.K., Slutzki, G.: Independence numbers of product graphs. Appl. Math. Lett. 7(4), 91\u201394 (1994)","journal-title":"Appl. Math. Lett."},{"key":"2600_CR13","first-page":"173","volume":"74","author":"Sandi Klav\u017ear","year":"2005","unstructured":"Klav\u017ear, Sandi: Some new bounds and exact results on the independence number of Cartesian product graphs. Ars Combin. 74, 173\u2013186 (2005)","journal-title":"Ars Combin."},{"issue":"5","key":"2600_CR14","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s00373-010-0997-7","volume":"27","author":"Sandi Klav\u017ear","year":"2011","unstructured":"Klav\u017ear, Sandi, Meki\u0161, Ga\u0161per: On idomatic partitions of direct products of complete graphs. Graphs Combin. 27(5), 713\u2013726 (2011)","journal-title":"Graphs Combin."},{"issue":"2","key":"2600_CR15","doi-asserted-by":"publisher","first-page":"627","DOI":"10.7151\/dmgt.2296","volume":"42","author":"Kirsti Kuenzel","year":"2022","unstructured":"Kuenzel, Kirsti, Rall, Douglas F.: On well-covered direct products. Discuss. Math. Graph Theory 42(2), 627\u2013640 (2022)","journal-title":"Discuss. Math. Graph Theory"},{"issue":"1","key":"2600_CR16","doi-asserted-by":"publisher","first-page":"53","DOI":"10.7151\/dmgt.1023","volume":"16","author":"Richard J Nowakowski","year":"1996","unstructured":"Nowakowski, Richard J., Rall, Douglas F.: Associative graph products and their independence, domination and coloring numbers. Discuss. Math. Graph Theory 16(1), 53\u201379 (1996)","journal-title":"Discuss. Math. Graph Theory"},{"key":"2600_CR17","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/j.jctb.2014.02.010","volume":"108","author":"\u00c1gnes T\u00f3th","year":"2014","unstructured":"T\u00f3th, \u00c1gnes.: On the ultimate categorical independence ratio. J. Combin. Theory Ser. B 108, 29\u201339 (2014)","journal-title":"J. Combin. Theory Ser. B"},{"key":"2600_CR18","first-page":"199","volume":"33","author":"J Topp","year":"1992","unstructured":"Topp, J., Volkmann, L.: On the well-coveredness of products of graphs. Ars Comb. 33, 199\u2013215 (1992)","journal-title":"Ars Comb."}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-022-02600-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-022-02600-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-022-02600-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T21:45:51Z","timestamp":1677015951000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-022-02600-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,27]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["2600"],"URL":"https:\/\/doi.org\/10.1007\/s00373-022-02600-0","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"value":"0911-0119","type":"print"},{"value":"1435-5914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,27]]},"assertion":[{"value":"25 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 August 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 December 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Financial or non-financial interests"}},{"value":"Data sharing not applicable to this article as no datasets were generated or analysed during the current study.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Data sharing"}}],"article-number":"7"}}