{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T05:10:35Z","timestamp":1696482635687},"reference-count":9,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2014,6,23]],"date-time":"2014-06-23T00:00:00Z","timestamp":1403481600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"name":"the National Natural Science Foundation for Young Scientists of China","award":["(11201273) (61202017)"]},{"name":"the Natural Science Foundation for Young Scientists of Shanxi Province, China","award":["(2011021004) (2013021001-5)"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[2015,5]]},"abstract":"Abstract<\/jats:title>Let D<\/jats:italic> be a digraph with vertex set and arc set . A vertex x<\/jats:italic> is a k<\/jats:italic>\u2010king of D<\/jats:italic>, if for every , there is an \u2010path of length at most k<\/jats:italic>. A subset N<\/jats:italic> of is k<\/jats:italic>\u2010independent if for every pair of vertices , we have and ; it is l<\/jats:italic>\u2010absorbent if for every there exists such that . A \u2010kernel of D<\/jats:italic> is a k<\/jats:italic>\u2010independent and l<\/jats:italic>\u2010absorbent subset of . A k<\/jats:italic>\u2010kernel is a \u2010kernel. A digraph D<\/jats:italic> is k<\/jats:italic>\u2010quasitransitive, if for any path of length k<\/jats:italic>, x<\/jats:italic>0<\/jats:sub> and are adjacent. In this article, we will prove that a k<\/jats:italic>\u2010quasitransitive digraph with has a k<\/jats:italic>\u2010king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended \u2010cycle where each has at least two vertices. Using this fact, we show that every strong k<\/jats:italic>\u2010quasitransitive digraph has a \u2010kernel.<\/jats:p>","DOI":"10.1002\/jgt.21814","type":"journal-article","created":{"date-parts":[[2014,6,24]],"date-time":"2014-06-24T12:34:39Z","timestamp":1403613279000},"page":"55-62","source":"Crossref","is-referenced-by-count":5,"title":["k<\/i>\u2010Kings in k<\/i>\u2010Quasitransitive Digraphs"],"prefix":"10.1002","volume":"79","author":[{"given":"Ruixia","family":"Wang","sequence":"first","affiliation":[{"name":"SCHOOL OF MATHEMATICAL SCIENCE OF SHANXI UNIVERSITY TAIYUAN P.R. CHINA"}]},{"given":"Wei","family":"Meng","sequence":"additional","affiliation":[{"name":"SCHOOL OF MATHEMATICAL SCIENCE OF SHANXI UNIVERSITY TAIYUAN P.R. CHINA"}]}],"member":"311","published-online":{"date-parts":[[2014,6,23]]},"reference":[{"key":"e_1_2_4_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(97)00179-9"},{"key":"e_1_2_4_3_1","volume-title":"Digraphs: Theory, Algorithms and Applications","author":"Bang\u2010Jensen J.","year":"2000"},{"key":"e_1_2_4_4_1","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(200003)33:3<177::AID-JGT8>3.0.CO;2-1"},{"key":"e_1_2_4_5_1","doi-asserted-by":"publisher","DOI":"10.7151\/dmgt.1546"},{"key":"e_1_2_4_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2012.05.005"},{"key":"e_1_2_4_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2013.08.007"},{"key":"e_1_2_4_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(94)00169-J"},{"key":"e_1_2_4_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02478336"},{"key":"e_1_2_4_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20426"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.21814","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.21814","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T03:00:21Z","timestamp":1696388421000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.21814"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,23]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,5]]}},"alternative-id":["10.1002\/jgt.21814"],"URL":"https:\/\/doi.org\/10.1002\/jgt.21814","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,23]]}}}
