{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:39:23Z","timestamp":1740155963210,"version":"3.37.3"},"reference-count":8,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2020,10]]},"abstract":"<jats:p>Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of [Formula: see text] is called the isolate domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we show that for every proper interval graph [Formula: see text], [Formula: see text]. Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci. 22(2) (2016) 232\u2013241].<\/jats:p>","DOI":"10.1142\/s1793830920500652","type":"journal-article","created":{"date-parts":[[2020,6,20]],"date-time":"2020-06-20T10:48:23Z","timestamp":1592650103000},"page":"2050065","source":"Crossref","is-referenced-by-count":2,"title":["Characterization of some classes of graphs with equal domination number and isolate domination number"],"prefix":"10.1142","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8883-8312","authenticated-orcid":false,"given":"Davood","family":"Bakhshesh","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Bojnord, Bojnord, Iran"}]}],"member":"219","published-online":{"date-parts":[[2020,6,20]]},"reference":[{"issue":"31","key":"S1793830920500652BIB001","first-page":"1543","volume":"9","author":"Arriola B. H.","year":"2015","journal-title":"Appl. Math. Sci."},{"key":"S1793830920500652BIB002","doi-asserted-by":"publisher","DOI":"10.26708\/IJMSC.2013.3.3.11"},{"issue":"2","key":"S1793830920500652BIB003","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.ajmsc.2015.10.001","volume":"22","author":"Hamid I. S.","year":"2016","journal-title":"Arab J. Math. Sci."},{"key":"S1793830920500652BIB004","doi-asserted-by":"publisher","DOI":"10.5614\/ejgta.2016.4.1.8"},{"key":"S1793830920500652BIB005","first-page":"127","volume":"3","author":"Hamid S.","year":"2013","journal-title":"Bull. Int. Math. Virtual Inst."},{"volume-title":"Fundamentals of Domination in Graphs","year":"1998","author":"Haynes T. W.","key":"S1793830920500652BIB006"},{"key":"S1793830920500652BIB007","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(93)90308-I"},{"key":"S1793830920500652BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/j.akcej.2017.01.003"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830920500652","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T01:47:27Z","timestamp":1723081647000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830920500652"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,20]]},"references-count":8,"journal-issue":{"issue":"05","published-print":{"date-parts":[[2020,10]]}},"alternative-id":["10.1142\/S1793830920500652"],"URL":"https:\/\/doi.org\/10.1142\/s1793830920500652","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"type":"print","value":"1793-8309"},{"type":"electronic","value":"1793-8317"}],"subject":[],"published":{"date-parts":[[2020,6,20]]}}}