{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:40:59Z","timestamp":1760240459584,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,24]],"date-time":"2019-06-24T00:00:00Z","timestamp":1561334400000},"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>Given a graph     G = ( V , E )    , a function     f : V \u2192 { 0 , 1 , 2 , \u22ef }     is said to be a total dominating function if      \u2211  u \u2208 N ( v )   f  ( u )  &gt; 0     for every     v \u2208 V    , where     N ( v )     denotes the open neighbourhood of v. Let      V i  =  { x \u2208 V :  f  ( x )  = i }     . We say that a function     f : V \u2192 { 0 , 1 , 2 }     is a total weak Roman dominating function if f is a total dominating function and for every vertex     v \u2208  V 0      there exists     u \u2208 N  ( v )  \u2229  (  V 1  \u222a  V 2  )      such that the function     f \u2032    , defined by      f \u2032   ( v )  = 1    ,      f \u2032   ( u )  = f  ( u )  \u2212 1     and      f \u2032   ( x )  = f  ( x )      whenever     x \u2208 V \\ { u , v }    , is a total dominating function as well. The weight of a function f is defined to be     w  ( f )  =  \u2211  v \u2208 V   f  ( v )  .     In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by      \u03b3  t r    ( G )     , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on      \u03b3  t r    ( G )      and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.<\/jats:p>","DOI":"10.3390\/sym11060831","type":"journal-article","created":{"date-parts":[[2019,6,24]],"date-time":"2019-06-24T11:01:57Z","timestamp":1561374117000},"page":"831","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Total Weak Roman Domination in Graphs"],"prefix":"10.3390","volume":"11","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"}]},{"given":"Luis P.","family":"Montejano","sequence":"additional","affiliation":[{"name":"CONACYT Research Fellow\u2014Centro de Investigaci\u00f3n en Matem\u00e1ticas, 36023 Guanajuato, GTO, Mexico"}],"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":[[2019,6,24]]},"reference":[{"key":"ref_1","first-page":"19","article-title":"Protection of a graph","volume":"67","author":"Cockayne","year":"2005","journal-title":"Util. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1038\/scientificamerican1299-136","article-title":"Defend the Roman Empire!","volume":"281","author":"Stewart","year":"1999","journal-title":"Sci. Am."},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1007\/s10878-012-9482-y","article-title":"Roman domination on strongly chordal graphs","volume":"26","author":"Liu","year":"2013","journal-title":"J. Comb. Optim."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"501","DOI":"10.2298\/AADM160802017A","article-title":"Total roman domination in graphs","volume":"10","author":"Henning","year":"2016","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/S0012-365X(02)00811-7","article-title":"Defending the Roman Empire\u2014A new strategy","volume":"266","author":"Henning","year":"2003","journal-title":"Discret. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.dam.2014.06.016","article-title":"Bounds on weak roman and 2-rainbow domination numbers","volume":"178","author":"Chellali","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"ref_8","first-page":"87","article-title":"Secure domination, weak Roman domination and forbidden subgraphs","volume":"39","author":"Cockayne","year":"2003","journal-title":"Bull. Inst. Combin. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.dam.2018.03.039","article-title":"On the weak Roman domination number of lexicographic product graphs","volume":"263","author":"Valveny","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"319","DOI":"10.2298\/FIL1901319V","article-title":"Protection of graphs with emphasis on cartesian product graphs","volume":"33","author":"Valveny","year":"2019","journal-title":"Filomat"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1016\/j.ipl.2015.05.006","article-title":"On secure domination in graphs","volume":"115","author":"Chellali","year":"2015","journal-title":"Inform. Process. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"163","DOI":"10.2989\/QM.2008.31.2.5.477","article-title":"Vertex covers and secure domination in graphs","volume":"31","author":"Burger","year":"2008","journal-title":"Quaest. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"267","DOI":"10.7151\/dmgt.1405","article-title":"Secure domination and secure total domination in graphs","volume":"28","author":"Klostermeyer","year":"2008","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_14","first-page":"247","article-title":"Secure total domination in graphs","volume":"74","author":"Benecke","year":"2007","journal-title":"Util. Math."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"225","DOI":"10.2140\/pjm.1975.61.225","article-title":"Relations between packing and covering numbers of a tree","volume":"61","author":"Meir","year":"1975","journal-title":"Pacific J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/jgt.10027","article-title":"Domination in planar graphs with small diameter","volume":"40","author":"Goddard","year":"2002","journal-title":"J. Graph Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.dam.2017.11.019","article-title":"On the (adjacency) metric dimension of corona and strong product graphs and their local variants: Combinatorial and computational results","volume":"236","author":"Fernau","year":"2018","journal-title":"Discret. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Fernau, H., and Rodr\u00edguez-Vel\u00e1zquez, J.A. (2014). Notions of Metric Dimension of Corona Products: Combinatorial and Computational Results, Springer International Publishing.","DOI":"10.1007\/978-3-319-06686-8_12"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.dam.2018.03.082","article-title":"On the super domination number of lexicographic product graphs","volume":"263","author":"Dettlaff","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"ref_21","unstructured":"Garey, M.R., and Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman & Co."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/831\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:00:53Z","timestamp":1760187653000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/831"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,24]]},"references-count":21,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["sym11060831"],"URL":"https:\/\/doi.org\/10.3390\/sym11060831","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,6,24]]}}}