Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21288320230901Total restrained Roman domination5755871442610.22049/cco.2022.27628.1303ENJafar AmjadiAzarbaijan Shahid Madani University0000-0001-9340-4773Babak SamadiFaculty of Mathematical Sciences, Alzahra University, Tehran, IranLutz VolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20220114Let $G$ be a graph with vertex set $V(G)$. A Roman dominating function (RDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to a vertex $u$ with $f(u)=2$. If $f$ is an RDF on $G$, then let $V_i=\{v\in V(G): f(v)=i\}$ for $i\in\{0,1,2\}$. An RDF $f$ is called a restrained (total) Roman dominating function if the subgraph induced by $V_0$ (induced by $V_1\cup V_2$) has no isolated vertex. A total and restrained Roman dominating function is a total restrained Roman dominating function. The total restrained Roman domination number $\gamma_{trR}(G)$ on a graph $G$ is the minimum weight of a total restrained Roman dominating function on the graph $G$. We initiate the study of total restrained Roman domination number and present several sharp bounds on $\gamma_{trR}G)$. In addition, we determine this parameter for some classes of graphs.https://comb-opt.azaruniv.ac.ir/article_14426_d5f2a14f23334b09e3f71e675a41e54c.pdf