{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:22:14Z","timestamp":1760239334622,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,9]],"date-time":"2020-11-09T00:00:00Z","timestamp":1604880000000},"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>Let G be a graph with no isolated vertex and f:V(G)\u2192{0,1,2} a function. Let Vi={v\u2208V(G):f(v)=i} for every i\u2208{0,1,2}. The function f is an outer-independent Roman dominating function on G if V0 is an independent set and every vertex in V0 is adjacent to at least one vertex in V2. The minimum weight \u03c9(f)=\u2211v\u2208V(G)f(v) among all outer-independent Roman dominating functions f on G is the outer-independent Roman domination number of G. This paper is devoted to the study of the outer-independent Roman domination number of a graph, and it is a contribution to the special issue \u201cTheoretical Computer Science and Discrete Mathematics\u201d of Symmetry. In particular, we obtain new tight bounds for this parameter, and some of them improve some well-known results. We also provide closed formulas for the outer-independent Roman domination number of rooted product graphs.<\/jats:p>","DOI":"10.3390\/sym12111846","type":"journal-article","created":{"date-parts":[[2020,11,10]],"date-time":"2020-11-10T21:47:11Z","timestamp":1605044831000},"page":"1846","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On the Outer-Independent Roman Domination in Graphs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2806-4842","authenticated-orcid":false,"given":"Abel Cabrera","family":"Mart\u00ednez","sequence":"first","affiliation":[{"name":"Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain"}]},{"given":"Suitberto Cabrera","family":"Garc\u00eda","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica e Investigaci\u00f3n Operativa Aplicadas y Calidad, Universitat Polit\u00e9cnica de Valencia, Camino de Vera s\/n, 46022 Valencia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0953-2500","authenticated-orcid":false,"given":"Andr\u00e9s","family":"Carri\u00f3n Garc\u00eda","sequence":"additional","affiliation":[{"name":"Centro de Gesti\u00f3n de la Calidad y del Cambio, Universitat Polit\u00e9cnica de Valencia, Camino de Vera s\/n, 46022 Valencia, Spain"}]},{"given":"Angela Mar\u00eda","family":"Grisales del Rio","sequence":"additional","affiliation":[{"name":"Centro de Gesti\u00f3n de la Calidad y del Cambio, Universitat Polit\u00e9cnica de Valencia, Camino de Vera s\/n, 46022 Valencia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Fundamentals of Domination in Graphs, Marcel Dekker, Inc.","DOI":"10.1002\/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Domination in Graphs: Advanced Topics, Marcel Dekker, Inc.","DOI":"10.1002\/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F"},{"key":"ref_3","first-page":"133","article-title":"\u00dcber extreme Punkt-und Kantenmengen","volume":"2","author":"Gallai","year":"1959","journal-title":"Ann. Univ. Sci. Budapestinensis Rolando E\u00f6tv\u00f6s Nomin. Sect. Math."},{"key":"ref_4","unstructured":"Berge, C. (1962). Theory of Graphs and Its Applications, CRC Press."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ore, O. (1962). Theory of Graphs, American Mathematical Society.","DOI":"10.1090\/coll\/038"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/j.disc.2012.11.031","article-title":"Independent domination in graphs: A survey and recent results","volume":"313","author":"Goddard","year":"2013","journal-title":"Discret. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.disc.2003.06.004","article-title":"Roman domination in graphs","volume":"278","author":"Cockayne","year":"2004","journal-title":"Discret. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2547","DOI":"10.1080\/00207160.2017.1301437","article-title":"Outer independent Roman dominating functions in graphs","volume":"94","author":"Chellali","year":"2017","journal-title":"Int. J. Comput. Math."},{"key":"ref_9","first-page":"267","article-title":"A constructive characterization of vertex cover Roman trees","volume":"41","author":"Kuziak","year":"2018","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"35544","DOI":"10.1109\/ACCESS.2018.2851972","article-title":"Outer-independent Roman domination number of tree","volume":"6","author":"Dehgardi","year":"2018","journal-title":"IEEE Access"},{"key":"ref_11","first-page":"185","article-title":"On trees with equal Roman domination and outer-independent Roman domination numbers","volume":"4","author":"Sheikholeslami","year":"2019","journal-title":"Commun. Comb. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0004972700007760","article-title":"A new graph product and its spectrum","volume":"18","author":"Godsil","year":"1978","journal-title":"Bull. Aust. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1846\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:30:57Z","timestamp":1760178657000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1846"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,9]]},"references-count":12,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["sym12111846"],"URL":"https:\/\/doi.org\/10.3390\/sym12111846","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,11,9]]}}}