{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T11:47:24Z","timestamp":1780919244261,"version":"3.54.1"},"reference-count":13,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,9]],"date-time":"2020-02-09T00:00:00Z","timestamp":1581206400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>For a graph     G = ( V , E )     with vertex set     V = V ( G )     and edge set     E = E ( G )    , a Roman     { 3 }    -dominating function (R    { 3 }    -DF) is a function     f : V ( G ) \u2192 { 0 , 1 , 2 , 3 }     having the property that      \u2211  u \u2208  N G   ( v )    f  ( u )  \u2265 3    , if     f ( v ) = 0    , and      \u2211  u \u2208  N G   ( v )    f  ( u )  \u2265 2    , if     f ( v ) = 1     for any vertex     v \u2208 V ( G )    . The weight of a Roman     { 3 }    -dominating function f is the sum     f  ( V )  =  \u2211  v \u2208 V ( G )   f  ( v )      and the minimum weight of a Roman     { 3 }    -dominating function on G is the Roman     { 3 }    -domination number of G, denoted by      \u03b3  { R 3 }    ( G )     . Let G be a graph with no isolated vertices. The total Roman     { 3 }    -dominating function on G is an R    { 3 }    -DF f on G with the additional property that every vertex     v \u2208 V     with     f ( v ) \u2260 0     has a neighbor w with     f ( w ) \u2260 0    . The minimum weight of a total Roman     { 3 }    -dominating function on G, is called the total Roman     { 3 }    -domination number denoted by      \u03b3  t { R 3 }    ( G )     . We initiate the study of total Roman     { 3 }    -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman     { 3 }    -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman     { 3 }    -domination for bipartite graphs.<\/jats:p>","DOI":"10.3390\/sym12020268","type":"journal-article","created":{"date-parts":[[2020,2,10]],"date-time":"2020-02-10T11:48:51Z","timestamp":1581335331000},"page":"268","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Total Roman {3}-domination in Graphs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0764-4135","authenticated-orcid":false,"given":"Zehui","family":"Shao","sequence":"first","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9373-3390","authenticated-orcid":false,"given":"Doost Ali","family":"Mojdeh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lutz","family":"Volkmann","sequence":"additional","affiliation":[{"name":"Lehrstuhl II f\u00fcr Mathematik, RWTH Aachen University, 52056 Aachen, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,9]]},"reference":[{"key":"ref_1","first-page":"25","article-title":"On connected (\u03b3,k)-critical graphs","volume":"46","author":"Mojdeh","year":"2010","journal-title":"Australas. 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Roman {3}-domination (Double Italian Domination). article in press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/268\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:56:16Z","timestamp":1760172976000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/268"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,9]]},"references-count":13,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["sym12020268"],"URL":"https:\/\/doi.org\/10.3390\/sym12020268","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,9]]}}}