%0 Journal Article
%T Total restrained Roman domination
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Amjadi, Jafar
%A Samadi, Babak
%A Volkmann, Lutz
%D 2023
%\ 09/01/2023
%V 8
%N 3
%P 575-587
%! Total restrained Roman domination
%K Total restrained domination
%K total restrained Roman domination
%K total restrained Roman domination number
%R 10.22049/cco.2022.27628.1303
%X Let $G$ 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.
%U https://comb-opt.azaruniv.ac.ir/article_14426_d5f2a14f23334b09e3f71e675a41e54c.pdf