{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:30:03Z","timestamp":1747189803409,"version":"3.40.5"},"reference-count":11,"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":[[2021,2]]},"abstract":"<jats:p> A set [Formula: see text] is called an open detour set of [Formula: see text] if for each vertex [Formula: see text] in [Formula: see text], either (1) [Formula: see text] is a detour simplicial vertex of [Formula: see text] and [Formula: see text] or (2) [Formula: see text] is an internal vertex of an [Formula: see text]-[Formula: see text] detour for some [Formula: see text]. An open detour set of minimum cardinality is called a minimum open detour set and this cardinality is the open detour number of [Formula: see text], denoted by [Formula: see text]. Connected graphs of order [Formula: see text] with open detour number [Formula: see text] or [Formula: see text] are characterized. It is shown that for any two positive 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], where [Formula: see text] is the detour number of [Formula: see text]. It is also shown that for every pair of positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the open geodetic number of [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s1793830920500883","type":"journal-article","created":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T13:01:31Z","timestamp":1594645291000},"page":"2050088","source":"Crossref","is-referenced-by-count":1,"title":["The open detour number of a graph"],"prefix":"10.1142","volume":"13","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":"V. R.","family":"Sunil Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rajalakshmi Engineering College, Thandalam 602 105, India"}]}],"member":"219","published-online":{"date-parts":[[2020,8,31]]},"reference":[{"volume-title":"Distance in Graphs","year":"1990","author":"Buckley F.","key":"S1793830920500883BIB001"},{"key":"S1793830920500883BIB002","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/j.disc.2006.07.019","volume":"307","author":"Bullock F.","year":"2007","journal-title":"Discrete Math."},{"key":"S1793830920500883BIB003","first-page":"51","volume":"31","author":"Chartrand G.","year":"2001","journal-title":"Bull. ICA"},{"key":"S1793830920500883BIB004","first-page":"97","volume":"64","author":"Chartrand G.","year":"2003","journal-title":"Util. 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