{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T18:41:38Z","timestamp":1781030498064,"version":"3.54.1"},"reference-count":11,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2011,12]]},"abstract":"<jats:p> For a commutative ring R, let Z(R) be its set of zero-divisors. The total graph of R, denoted by T<jats:sub>\u0393<\/jats:sub>(R), is the undirected graph with vertex set R, and for distinct x, y \u2208 R, the vertices x and y are adjacent if and only if x + y \u2208 Z(R). Tamizh Chelvam and Asir studied about the domination in the total graph of a commutative ring R. In particular, it was proved that the domination number \u03b3(T<jats:sub>\u0393<\/jats:sub>(\u2124<jats:sub>n<\/jats:sub>)) = p<jats:sub>1<\/jats:sub> where p<jats:sub>1<\/jats:sub> is the smallest prime divisor of n. In this paper, we characterize all the \u03b3-sets in T<jats:sub>\u0393<\/jats:sub>(\u2124<jats:sub>n<\/jats:sub>). Also, we obtain the values of other domination parameters like independent, total and perfect domination numbers of the total graph on \u2124<jats:sub>n<\/jats:sub>. <\/jats:p>","DOI":"10.1142\/s1793830911001309","type":"journal-article","created":{"date-parts":[[2012,1,4]],"date-time":"2012-01-04T14:00:28Z","timestamp":1325685628000},"page":"413-421","source":"Crossref","is-referenced-by-count":13,"title":["DOMINATION IN THE TOTAL GRAPH ON \u2124<sub>n<\/sub>"],"prefix":"10.1142","volume":"03","author":[{"given":"T. TAMIZH","family":"CHELVAM","sequence":"first","affiliation":[{"name":"Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T.","family":"ASIR","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2012,4,5]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2009.03.013"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2008.06.028"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1998.7840"},{"key":"rf4","volume-title":"Introduction to Graph Theory","author":"Chartrand G.","year":"2006"},{"key":"rf5","volume-title":"Abstract Algebra","author":"Dummit D. S.","year":"2008"},{"key":"rf6","volume-title":"Fundamental of Domination in Graphs","author":"Haynes T. W.","year":"1998"},{"key":"rf7","volume-title":"Commutative Rings with Zero Divisors","author":"Huckaba J. A.","year":"1988"},{"key":"rf8","volume-title":"Algebra","author":"Hungerford T. W.","year":"2005"},{"key":"rf9","volume-title":"Commutative Rings","author":"Kaplansky I.","year":"1974"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2006.01.019"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2006.07.025"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830911001309","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T21:40:33Z","timestamp":1565127633000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830911001309"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12]]},"references-count":11,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2012,4,5]]},"published-print":{"date-parts":[[2011,12]]}},"alternative-id":["10.1142\/S1793830911001309"],"URL":"https:\/\/doi.org\/10.1142\/s1793830911001309","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,12]]}}}