{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:07:39Z","timestamp":1753880859714,"version":"3.41.2"},"reference-count":13,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"crossref","award":["12061059"],"award-info":[{"award-number":["12061059"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Inter. Net."],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:p> The concept of [Formula: see text]-independent set, introduced by Fink and Jacobson in 1986, is a natural generalization of classical independence set. A [Formula: see text]-independent set is a set of vertices whose induced subgraph has maximum degree at most [Formula: see text]. The [Formula: see text]-independence number of [Formula: see text], denoted by [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-independent set of [Formula: see text]. As a natural counterpart of the [Formula: see text]-independence number, we introduced the concept of [Formula: see text]-edge-independence number. An edge set [Formula: see text] in [Formula: see text] is called [Formula: see text]-edge-independent if the maximum degree of the subgraph induced by the edges in [Formula: see text] is less or equal to [Formula: see text]. The [Formula: see text]-edge-independence number, denoted [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-edge-independent set. In this paper, we study the Nordhaus\u2013Gaddum-type results for the parameter [Formula: see text] and [Formula: see text]. We obtain sharp upper and lower bounds of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. Some graph classes attaining these bounds are also given. <\/jats:p>","DOI":"10.1142\/s021926592350007x","type":"journal-article","created":{"date-parts":[[2023,6,22]],"date-time":"2023-06-22T08:53:41Z","timestamp":1687424021000},"source":"Crossref","is-referenced-by-count":1,"title":["Nordhaus\u2013Gaddum-Type Results for the k-Independent Number of Graphs"],"prefix":"10.1142","volume":"24","author":[{"given":"Zhao","family":"Wang","sequence":"first","affiliation":[{"name":"College of Science, China Jiliang University, Hangzhou 310018, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongfang","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Education, Qinghai Normal University, Xining, Qinghai 810008, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuhu","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Southwest Jiaotong University, Chengdu 610031, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2024,3,22]]},"reference":[{"key":"S021926592350007XBIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2011.12.018"},{"key":"S021926592350007XBIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-970-5"},{"key":"S021926592350007XBIB003","doi-asserted-by":"publisher","DOI":"10.1111\/j.2164-0947.1974.tb01571.x"},{"key":"S021926592350007XBIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-011-1040-3"},{"key":"S021926592350007XBIB005","doi-asserted-by":"publisher","DOI":"10.37236\/2646"},{"key":"S021926592350007XBIB006","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190150110"},{"key":"S021926592350007XBIB007","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2013.02.008"},{"key":"S021926592350007XBIB008","first-page":"283","volume-title":"Graph Theory with Applications to Algorithms and Computer Science","volume":"68","author":"Fink J. F.","year":"1985"},{"key":"S021926592350007XBIB009","first-page":"301","volume-title":"Graph Theory with Applications to Algorithms and Computer Science","author":"Fink J. F.","year":"1985"},{"key":"S021926592350007XBIB010","doi-asserted-by":"publisher","DOI":"10.1007\/s10878-014-9783-4"},{"key":"S021926592350007XBIB011","first-page":"#P01","volume":"1","author":"Mao Y.","year":"2018","journal-title":"Art Discrete Appl. 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