{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:08Z","timestamp":1740134528265,"version":"3.37.3"},"reference-count":16,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61309015","61173121"],"award-info":[{"award-number":["61309015","61173121"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61309015","61173121"],"award-info":[{"award-number":["61309015","61173121"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>Denote by<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>\u03b3<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>the domination number of a digraph<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u25a1<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>the Cartesian product of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>, the directed cycles of length<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>m<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn fontstyle=\"italic\">2<\/mml:mn><\/mml:math>. In 2010, Liu et al. determined the exact values of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>\u03b3<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u25a1<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2,3<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>4,5<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>6<\/mml:mn><\/mml:math>. In 2013, Mollard determined the exact values of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>\u03b3<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u25a1<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>3<\/mml:mn><mml:mi>k<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math>. In this paper, we give lower and upper bounds of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>\u03b3<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u25a1<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>3<\/mml:mn><mml:mi>k<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>for different cases. In particular,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mo stretchy=\"false\">\u2308<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn fontstyle=\"italic\">2<\/mml:mn><mml:mi>k<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn fontstyle=\"italic\">1<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mi>n<\/mml:mi><mml:mo>\/<\/mml:mo><mml:mn fontstyle=\"italic\">2<\/mml:mn><mml:mo stretchy=\"false\">\u2309<\/mml:mo><mml:mo>\u2264<\/mml:mo><mml:mi>\u03b3<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">3<\/mml:mn><mml:mi>k<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn fontstyle=\"italic\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>\u25a1<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2264<\/mml:mo><mml:mo stretchy=\"false\">\u230a<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mn fontstyle=\"italic\">2<\/mml:mn><mml:mi>k<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn fontstyle=\"italic\">1<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mi>n<\/mml:mi><mml:mo>\/<\/mml:mo><mml:mn fontstyle=\"italic\">2<\/mml:mn><mml:mo stretchy=\"false\">\u230b<\/mml:mo><mml:mo>+<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math>. Based on the established result, the exact values of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>\u03b3<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u25a1<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>are determined for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>7<\/mml:mn><\/mml:math>and 10 by the combination of the dynamic algorithm, and an upper bound for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mi>\u03b3<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">13<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>\u25a1<\/mml:mo><mml:mi>C<\/mml:mi><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>is provided.<\/jats:p>","DOI":"10.1155\/2013\/619695","type":"journal-article","created":{"date-parts":[[2013,12,19]],"date-time":"2013-12-19T21:03:26Z","timestamp":1387487006000},"page":"1-7","source":"Crossref","is-referenced-by-count":5,"title":["On the Domination Number of Cartesian Product of Two Directed Cycles"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0764-4135","authenticated-orcid":true,"given":"Zehui","family":"Shao","sequence":"first","affiliation":[{"name":"School of Information Science and Technology, Chengdu University, Chengdu 610106, China"},{"name":"Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu 610106, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enqiang","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fangnian","family":"Lang","sequence":"additional","affiliation":[{"name":"School of Information Science and Technology, Chengdu University, Chengdu 610106, China"},{"name":"Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu 610106, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"year":"2011","series-title":"Discrete Mathematics and Its Applications","key":"7"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/0165-4896(88)90041-8"},{"volume":"208","year":"1998","series-title":"Monographs and Textbooks in Pure and Applied 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