{"id":"https://openalex.org/W2074608221","doi":"https://doi.org/10.1155/2013/619695","title":"On the Domination Number of Cartesian Product of Two Directed Cycles","display_name":"On the Domination Number of Cartesian Product of Two Directed Cycles","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2074608221","doi":"https://doi.org/10.1155/2013/619695","mag":"2074608221"},"language":"lv","primary_location":{"id":"doi:10.1155/2013/619695","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/619695","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/619695.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2013/619695.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012636371","display_name":"Zehui Shao","orcid":"https://orcid.org/0000-0003-0764-4135"},"institutions":[{"id":"https://openalex.org/I24201400","display_name":"Chengdu University of Information Technology","ror":"https://ror.org/01yxwrh59","country_code":"CN","type":"education","lineage":["https://openalex.org/I24201400"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zehui Shao","raw_affiliation_strings":["Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu 610106, China","School of Information Science and Technology, Chengdu University, Chengdu 610106, China"],"raw_orcid":"https://orcid.org/0000-0003-0764-4135","affiliations":[{"raw_affiliation_string":"Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu 610106, China","institution_ids":[]},{"raw_affiliation_string":"School of Information Science and Technology, Chengdu University, Chengdu 610106, China","institution_ids":["https://openalex.org/I24201400"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004232829","display_name":"Enqiang Zhu","orcid":"https://orcid.org/0000-0002-5245-7905"},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Enqiang Zhu","raw_affiliation_strings":["School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5003134676","display_name":"Fangnian Lang","orcid":null},"institutions":[{"id":"https://openalex.org/I24201400","display_name":"Chengdu University of Information Technology","ror":"https://ror.org/01yxwrh59","country_code":"CN","type":"education","lineage":["https://openalex.org/I24201400"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Fangnian Lang","raw_affiliation_strings":["Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu 610106, China","School of Information Science and Technology, Chengdu University, Chengdu 610106, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu 610106, China","institution_ids":[]},{"raw_affiliation_string":"School of Information Science and Technology, Chengdu University, Chengdu 610106, China","institution_ids":["https://openalex.org/I24201400"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5003134676"],"corresponding_institution_ids":["https://openalex.org/I24201400"],"apc_list":{"value":1050,"currency":"USD","value_usd":1050},"apc_paid":{"value":1050,"currency":"USD","value_usd":1050},"fwci":0.0,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.11505687,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"2013","issue":null,"first_page":"1","last_page":"7"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.998199999332428,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.998199999332428,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9805999994277954,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9424999952316284,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5115348100662231},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.296819269657135}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5115348100662231},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.296819269657135}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2013/619695","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/619695","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/619695.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jam/1394808311","is_oa":false,"landing_page_url":"https://projecteuclid.org/euclid.jam/1394808311","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:RePEc:hin:jnljam:619695","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2013/619695.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:7eef2c44a2dd4062b53038b174fb71cf","is_oa":true,"landing_page_url":"https://doaj.org/article/7eef2c44a2dd4062b53038b174fb71cf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Applied Mathematics, Vol 2013 (2013)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2013/619695","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/619695","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/619695.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1231384226","display_name":"\u57fa\u4e8eClifford\u4ee3\u6570\u7684\u6a21\u5f0f\u8bc6\u522b\u7b97\u6cd5\u7814\u7a76","funder_award_id":"61173121","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5454197433","display_name":null,"funder_award_id":"61309015","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2074608221.pdf","grobid_xml":"https://content.openalex.org/works/W2074608221.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1493071618","https://openalex.org/W1597381944","https://openalex.org/W1967091137","https://openalex.org/W1992801013","https://openalex.org/W1994365837","https://openalex.org/W2000865051","https://openalex.org/W2023782877","https://openalex.org/W2036674522","https://openalex.org/W2070698547","https://openalex.org/W2075458544","https://openalex.org/W2077912884","https://openalex.org/W2101183555","https://openalex.org/W2490805901","https://openalex.org/W2962985910","https://openalex.org/W3048030522"],"related_works":["https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2544423928","https://openalex.org/W2052122378","https://openalex.org/W2053286651","https://openalex.org/W2181743346","https://openalex.org/W2187401768","https://openalex.org/W2181413294","https://openalex.org/W2989452537","https://openalex.org/W4313731559"],"abstract_inverted_index":{"Denote":[0],"by<mml:math":[1],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[2,11,13,18,20,27,40,44,54,58,70,74,80,107,113,128],"id=\"M1\"><mml:mi>\u03b3</mml:mi><mml:mo":[3],"stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo":[4],"stretchy=\"false\">)</mml:mo></mml:math>the":[5],"domination":[6],"number":[7],"of":[8,25,119],"a":[9],"digraph<mml:math":[10],"id=\"M2\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math>and<mml:math":[12],"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":[14],"Cartesian":[15],"product":[16],"of<mml:math":[17,39,53,69,106],"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":[19],"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>,":[21],"the":[22,36,50,100,103,117,120],"directed":[23],"cycles":[24],"length<mml:math":[26],"id=\"M6\"><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>\u2265</mml:mo><mml:mn":[28],"fontstyle=\"italic\">2</mml:mn></mml:math>.":[29],"In":[30,46,60,78],"2010,":[31],"Liu":[32],"et":[33],"al.":[34],"determined":[35,49,111],"exact":[37,51,104],"values":[38,52,105],"id=\"M7\"><mml:mi>\u03b3</mml:mi><mml:mo":[41],"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":[42,56,72,109],"stretchy=\"false\">)</mml:mo></mml:math>for<mml:math":[43,57],"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>.":[45],"2013,":[47],"Mollard":[48],"id=\"M9\"><mml:mi>\u03b3</mml:mi><mml:mo":[55],"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>.":[59],"this":[61],"paper,":[62],"we":[63],"give":[64],"lower":[65],"and":[66,123],"upper":[67,125],"bounds":[68],"id=\"M11\"><mml:mi>\u03b3</mml:mi><mml:mo":[71],"stretchy=\"false\">)</mml:mo></mml:math>with<mml:math":[73],"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":[75],"different":[76],"cases.":[77],"particular,<mml:math":[79],"id=\"M13\"><mml:mo":[81],"stretchy=\"false\">\u2308</mml:mo><mml:mfenced":[82],"separators=\"|\"><mml:mrow><mml:mn":[83,93],"fontstyle=\"italic\">2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn":[84,94],"fontstyle=\"italic\">1</mml:mn></mml:mrow></mml:mfenced><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mn":[85,95],"fontstyle=\"italic\">2</mml:mn><mml:mo":[86,96],"stretchy=\"false\">\u2309</mml:mo><mml:mo>\u2264</mml:mo><mml:mi>\u03b3</mml:mi><mml:mo":[87],"stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn":[88,130],"fontstyle=\"italic\">3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn":[89],"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":[90],"stretchy=\"false\">)</mml:mo><mml:mo>\u2264</mml:mo><mml:mo":[91],"stretchy=\"false\">\u230a</mml:mo><mml:mfenced":[92],"stretchy=\"false\">\u230b</mml:mo><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:math>.":[97],"Based":[98],"on":[99],"established":[101],"result,":[102],"id=\"M14\"><mml:mi>\u03b3</mml:mi><mml:mo":[108],"stretchy=\"false\">)</mml:mo></mml:math>are":[110],"for<mml:math":[112,127],"id=\"M15\"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:math>and":[114],"10":[115],"by":[116],"combination":[118],"dynamic":[121],"algorithm,":[122],"an":[124],"bound":[126],"id=\"M16\"><mml:mi>\u03b3</mml:mi><mml:mo":[129],"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":[131],"stretchy=\"false\">)</mml:mo></mml:math>is":[132],"provided.":[133]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-07-31T08:31:51.225901","created_date":"2025-10-10T00:00:00"}