{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T17:46:51Z","timestamp":1760550411145},"reference-count":20,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","funder":[{"name":"DST-INSPIRE FELLOWSHIP","award":["IF 110684"],"award-info":[{"award-number":["IF 110684"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:p> Let [Formula: see text] be a commutative ring with identity. The cozero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is [Formula: see text], the set of all non-zero and non-unit elements of [Formula: see text]. Two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The Crosscap of a graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that the graph can be embedded in the non-orientable surface [Formula: see text]. The planar graph is called [Formula: see text]-outerplanar if removing all the vertices incident on the outer face yields a [Formula: see text]-outerplanar. The Outerplanarity index of a graph [Formula: see text] is the smallest [Formula: see text] such that [Formula: see text] is [Formula: see text]-outerplanar. In this paper, we characterize the class of rings [Formula: see text] (up to isomorphism) for which [Formula: see text]. Further we characterize all finite rings [Formula: see text] (up to isomorphism) for which [Formula: see text] has an outerplanarity index two. <\/jats:p>","DOI":"10.1142\/s1793830917500744","type":"journal-article","created":{"date-parts":[[2017,11,14]],"date-time":"2017-11-14T02:59:45Z","timestamp":1510628385000},"page":"1750074","source":"Crossref","is-referenced-by-count":4,"title":["Rings whose cozero-divisor graph has crosscap number at most two"],"prefix":"10.1142","volume":"09","author":[{"given":"A.","family":"Mallika","sequence":"first","affiliation":[{"name":"Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, Tamilnadu, India"}]},{"given":"R.","family":"Kala","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, Tamilnadu, India"}]}],"member":"219","published-online":{"date-parts":[[2017,12,13]]},"reference":[{"key":"S1793830917500744BIB001","first-page":"753","volume":"35","author":"Afkhami M.","year":"2011","journal-title":"Southeast Asian Bull. Math."},{"issue":"4","key":"S1793830917500744BIB002","first-page":"935","volume":"35","author":"Afkhami M.","year":"2012","journal-title":"Bull. Malays. Math. Sci. Soc (2)"},{"key":"S1793830917500744BIB003","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498812501034"},{"key":"S1793830917500744BIB004","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498813501132"},{"key":"S1793830917500744BIB005","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498813500564"},{"key":"S1793830917500744BIB006","first-page":"61","volume-title":"Lecture Notes in Pure and Appl. Math.","volume":"220","author":"Anderson D. F.","year":"2001"},{"key":"S1793830917500744BIB007","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1998.7840"},{"key":"S1793830917500744BIB009","volume-title":"Introduction to Commutative Algebra","author":"Atiyah M. F.","year":"1969"},{"key":"S1793830917500744BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(88)90202-5"},{"key":"S1793830917500744BIB011","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190010114"},{"key":"S1793830917500744BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(78)90073-4"},{"key":"S1793830917500744BIB013","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2007.10.015"},{"key":"S1793830917500744BIB014","volume-title":"Graphs and Digraphs","author":"Chartrand G.","year":"1986"},{"key":"S1793830917500744BIB015","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-75520-3_33"},{"key":"S1793830917500744BIB016","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(79)90022-4"},{"key":"S1793830917500744BIB017","volume-title":"Commutative Rings.","author":"Kaplansky I.","year":"1974"},{"issue":"3","key":"S1793830917500744BIB018","first-page":"275","volume":"41","author":"Kulli V. R.","year":"1975","journal-title":"Proc. Indian Nat. Sci. Acad."},{"key":"S1793830917500744BIB019","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65759-7"},{"key":"S1793830917500744BIB020","first-page":"303","volume":"12","author":"Youngs J. W. T.","year":"1963","journal-title":"J. Math. Mech."},{"key":"S1793830917500744BIB021","series-title":"North-Holland Mathematics Studies 188","volume-title":"Graphs, Groups and Surfaces","author":"White T.","year":"1984"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830917500744","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T10:19:55Z","timestamp":1565173195000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830917500744"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12]]},"references-count":20,"journal-issue":{"issue":"06","published-online":{"date-parts":[[2017,12,13]]},"published-print":{"date-parts":[[2017,12]]}},"alternative-id":["10.1142\/S1793830917500744"],"URL":"https:\/\/doi.org\/10.1142\/s1793830917500744","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12]]}}}