{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:39:05Z","timestamp":1740155945337,"version":"3.37.3"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2020,2]]},"abstract":"<jats:p> A Steiner set [Formula: see text] of a connected graph [Formula: see text] of order [Formula: see text] is a restrained Steiner set if either [Formula: see text] or the subgraph [Formula: see text] has no isolated vertices. The minimum cardinality of a restrained Steiner set of [Formula: see text] is the restrained Steiner number of [Formula: see text], and is denoted by [Formula: see text]. A restrained Steiner set [Formula: see text] in a connected graph [Formula: see text] is called a minimal restrained Steiner set if no proper subset of [Formula: see text] is a restrained Steiner set of [Formula: see text]. The upper restrained Steiner number [Formula: see text] is the maximum cardinality of a minimal restrained Steiner set of [Formula: see text]. The upper restrained Steiner number of certain classes of graphs are determined. Connected graphs of order [Formula: see text] with upper restrained Steiner number [Formula: see text] or [Formula: see text] are characterized. It is shown that for every pair of integers [Formula: see text] and [Formula: see text], with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text]. Also, it is shown that for every pair of integers [Formula: see text] and [Formula: see text] with [Formula: see text] there exists a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper restrained geodetic number of the graph [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s1793830920500044","type":"journal-article","created":{"date-parts":[[2019,10,14]],"date-time":"2019-10-14T23:03:42Z","timestamp":1571094222000},"page":"2050004","source":"Crossref","is-referenced-by-count":5,"title":["The upper restrained Steiner number of a graph"],"prefix":"10.1142","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4715-5638","authenticated-orcid":false,"given":"J.","family":"John","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government College of Engineering, Tirunelveli 627 007, India"}]},{"given":"M. S.","family":"Malchijah Raj","sequence":"additional","affiliation":[{"name":"Research and Development Centre, Bharathiar University, Coimbatore 641 046, India"}]}],"member":"219","published-online":{"date-parts":[[2019,12,2]]},"reference":[{"key":"S1793830920500044BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s40840-014-0068-y"},{"volume-title":"Distance in Graphs","year":"1990","author":"Buckley F.","key":"S1793830920500044BIB002"},{"key":"S1793830920500044BIB003","first-page":"17","volume":"2","author":"Buckley F.","year":"1988","journal-title":"Scientia A"},{"key":"S1793830920500044BIB004","doi-asserted-by":"publisher","DOI":"10.7151\/dmgt.1112"},{"key":"S1793830920500044BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00456-8"},{"key":"S1793830920500044BIB006","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"S1793830920500044BIB007","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2004.08.039"},{"volume-title":"Steiner Distance in Graphs \u2014 A Survey","year":"2017","author":"Mao Y.","key":"S1793830920500044BIB009"},{"key":"S1793830920500044BIB011","first-page":"133","volume":"20","author":"Raines M.","year":"1999","journal-title":"Australas. 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