{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T20:18:04Z","timestamp":1774383484247,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T00:00:00Z","timestamp":1606089600000},"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>During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions.<\/jats:p>","DOI":"10.3390\/sym12111929","type":"journal-article","created":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T11:50:34Z","timestamp":1606132234000},"page":"1929","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Total Domination in Rooted Product Graphs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2806-4842","authenticated-orcid":false,"given":"Abel","family":"Cabrera 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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9082-7647","authenticated-orcid":false,"given":"Juan A.","family":"Rodr\u00edguez-Vel\u00e1zquez","sequence":"additional","affiliation":[{"name":"Departament d\u2019Enginyeria Inform\u00e0tica i Matem\u00e0tiques, Universitat Rovira i Virgili, Av. Pa\u00efsos Catalans 26, 43007 Tarragona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,23]]},"reference":[{"key":"ref_1","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Domination in Graphs: Advanced Topics., Marcel Dekker, Inc."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","unstructured":"Henning, M., and Yeo, A. (2013). Total Domination in Graphs. Springer Monographs in Mathematics, Springer.","DOI":"10.1007\/978-1-4614-6525-6"},{"key":"ref_4","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. Austral. Math. Soc."},{"key":"ref_5","first-page":"901","article-title":"Chemical Graphs Constructed from Rooted Product and Their Zagreb Indices","volume":"70","author":"Azari","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Cabrera Mart\u00ednez, A. (2020). Double outer-independent domination number of graphs. Quaest. Math.","DOI":"10.3390\/math8020194"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Cabrera Mart\u00ednez, A., Cabrera Garc\u00eda, S., Carri\u00f3n Garc\u00eda, A., and Hern\u00e1ndez Mira, F.A. (2020). Total Roman domination number of rooted product graphs. Mathematics, 8.","DOI":"10.3390\/math8101850"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cabrera Mart\u00ednez, A., Cabrera Garc\u00eda, S., Carri\u00f3n Garc\u00eda, A., and Grisales del Rio, A.M. (2020). On the outer-independent Roman domination in graphs. Symmetry, 12.","DOI":"10.3390\/sym12111846"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Cabrera Mart\u00ednez, A., Estrada-Moreno, A., and Rodr\u00edguez-Vel\u00e1zquez, J.A. (2020). Secure total domination in rooted product graphs. Mathematics, 8.","DOI":"10.3390\/math8040600"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Cabrera Mart\u00ednez, A., Montejano, L.P., and Rodr\u00edguez-Vel\u00e1zquez, J.A. (2019). 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