\documentclass[12pt,twoside,a4paper]{article}
\usepackage[english]{babel}
\usepackage{pdfsync}
\newcommand{\thepages}{81--93}
\newcommand{\doinumber}{007}

\usepackage{umj_8_2}            % Style package of Ural Mathematical Journal

\usepackage{amsmath}
\usepackage{tikz}
\usepackage{epsfig}         % If the package features are used
\usepackage{amsfonts,amssymb}   % If there is need to use math alphabets
                                % \mathbb{} - amsfonts.sty
                                % \mathfrak{} - eufrak.sty, amsfonts.sty
                                % symbols of font Euler - amssymb.sty
\newcommand{\Leftclt}{S. Kosari, S.M. Sheikholeslami, M. Chellali and  M. Hajjari}   % Left page header - name & family name of article author
\newcommand{\Rightclt}{Restrained Roman Reinforcement Number in Graphs} %% Right page header - Short title of article
\makeatletter\def\Hy@xspace@end{}\makeatother



%\usepackage{refcheck}
\setcounter{page}{81} %��������� � ...
\sloppy
\usepackage{srcltx}             % For easy move between WiNEdT and YAP

\begin{document}

\date{}

\Mainclt            % Page header print on the first page
% Enter article title and authors
% Reference to grants is a footnote to article title. You may use also Acknowledgments section for this purpose if you wish.

\begin{Titul}
{\large \bf \uppercase{Restrained Roman reinforcement number\linebreak in graphs}}\\[3ex]
{{\bf Saeed Kosari} \\[2ex] {\small Institute of Computing Science and Technology, Guangzhou University,\\

 Guangzhou 510006, China \\ \hrefemail{saeedkosari38@gzhu.edu.cn} }\\[3ex]
{\bf Seyed Mahmoud Sheikholeslami} \\[2ex] {\small Department of Mathematics, Azarbaijan Shahid Madani University,\\ Tabriz, I.R. Iran \\ \hrefemail{s.m.sheikholeslami@azaruniv.ac.ir} }\\[3ex]
{\bf Mustapha Chellali} \\[2ex] {\small LAMDA-RO Laboratory, Department of Mathematics,
University of Blida,\\ B.P. 270, Blida, Algeria \\ \hrefemail{m\_chellali@yahoo.com} }\\[3ex]
{\bf Maryam Hajjari} \\[2ex] {\small Department of Mathematics, Azarbaijan Shahid Madani University,\\ Tabriz, I.R. Iran \\ \hrefemail{m.hajjari260@gmail.com} }\\[5ex]}
\end{Titul}


\begin{Annot}
{\bf Abstract:}  A restrained Roman dominating function (RRD-function) on a graph $G=(V,E)$ is
a function $f$ from $V$ into $\{0,1,2\}$ satisfying: \textrm{(i) } every vertex
$u$ with $f(u)=0$ is adjacent to a vertex $v$ with $f(v)=2$; \textrm{(ii)} the
subgraph induced by the vertices assigned 0 under $f$ has no isolated
vertices. The weight of an RRD-function is the sum of its function value over
the whole set of vertices, and the restrained Roman domination number is the
minimum weight of an RRD-function on $G.$ In this paper, we begin the study of
the restrained Roman reinforcement number $r_{rR}(G)$ of a graph $G$ defined
as the cardinality of a smallest set of edges that we must add to the graph to
decrease its restrained Roman domination number. We first show
that the decision problem associated with the restrained Roman reinforcement
problem is NP-hard. Then several properties as well as some sharp bounds of
the restrained Roman reinforcement number are presented. In
particular it is established that $r_{rR}(T)=1$ for every tree
$T$ of order at least three.

{\bf Keywords:} Restrained Roman domination, Restrained Roman reinforcement.
\end{Annot}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %Please replace what follows by the body %%
 %of your article                         %%
                                          %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction} %This section must have a number.
\setcounter{equation}{0}
For definitions and notations not given here we refer {the reader
}to \cite{hhs}. We consider simple graphs~$G$ with vertex set $V=V(G)$ and
edge set $E=E(G)$. The \textit{order} of $G$ is $n=n(G)=|V|$. The \textit{open
neighborhood} of a vertex $v,$ denoted by $N(v)$ (or $N_{G}(v)$ to refer to $G$)
is the set \linebreak $\{u\in V(G)\mid uv\in E\}$ and its \textit{closed neighborhood} is
the set $N[v]=N_{G}[v]=N(v)\cup\{v\}$. The \textit{degree} of vertex $v\in V$
is $d(v)=d_{G}(v)=|N(v)|$. The \textit{maximum }and \textit{minimum degree} in
$G$ are denoted by $\Delta=\Delta(G)$ and $\delta=\delta(G)$, respectively. A
vertex of degree one is called a \textit{leaf} and its neighbor is called a
\textit{support vertex}. As usual, the \emph{path} (\emph{cycle},
\emph{complete }$p$\emph{-partite graph}, respectively) of order $n$ is
denoted by $P_{n}$ ($C_{n}$, $K_{n_{1},n_{2},\ldots,n_{p}}$, respectively). A
\emph{star }of order $n\geq2$ {is the graph }$K_{1,n-1}.$
%A tree $T$ is a \emph{double
%star} if it contains exactly two vertices that are not leaves. A double star
%with respectively $p$ and $q$ leaves attached at each support vertex is
%denoted by $DS_{p,q}.$
For a subset $S\subseteq V$, the subgraph induced by $S$ in $G$ is denoted as
$G[S]$.

A subset $S\subseteq V$ is a \emph{dominating set} of $G$ if every vertex in
$V\setminus S$ has a neighbor in $S$. The \emph{domination number }$\gamma(G)$
is the minimum cardinality of a dominating set of $G$.

As an application, in the design of networks for example, it is essential to study the effect
 of some modifications of the graph parameters on its structure. These modifications can be deletion or addition of vertices,
deletion or addition of edges. We refer the reader to chapter 7 of \cite{hhs} when the graph parameter is the domination number. The
\emph{reinforcement number} $r(G)$ of a graph~$G$ is the minimum
number of edges that have to be added to the graph $G$ in order to decrease
the domination number.  {Of course for graphs~$G$ with domination
number one it }was assumed that $r(G)=0.$   The
concept of the reinforcement number was introduced in 1990 by Kok and Mynhardt~\cite{KM}, and since then it has been defined and studied for
several other domination parameters, such as Roman domination~\cite{JS}, total
Roman domination~\cite{AAC}, quasi-total Roman domination~\cite{EACS}, Italian domination~\cite{HSS}, double Roman
domination~\cite{amjadi} and rainbow domination~\cite{ADF,SAKS}.

In 2015, Leely Pushpam and Padmapriea~\cite{PP}
{introduced the }concept of restrained Roman domination as a new
variation of Roman domination. A \textit{restrained Roman dominating function}
(RRD-function, for short) on a graph $G$ is a function $f:V\longrightarrow
\{0,1,2\}$ having the properties that \textrm{(i) }every vertex $u$ with
$f(u)=0$ is adjacent to a vertex $v$ with $f(v)=2$; and \textrm{(ii)} the
subgraph induced by the vertices assigned 0 under $f$ has no isolated
vertices. The weight of an RRD-function~$f$ is the sum $$w(f)=\sum_{v\in
V(G)}f(v)$$ and {the \emph{restrained Roman domination number} of $G$ denoted by
$\gamma_{rR}(G)$, is the minimum weight of an RRD-function on $G.$} Any
RRD-function~$f$ on $G$ can simply be referred  as $f=(V_{0},V_{1},V_{2})$,
where $V_{i}=\{v\in V(G):f(v)=i\}$ for $i\in\{0,1,2\}$. For
further studies on restrained Roman domination and its variants, see~\cite{AAC1,samadi,AAC2,V1,V2}.

In this paper, we are interested in starting the study of the
\emph{restrained Roman reinforcement} \emph{number} $r_{rR}(G)$ of a graph $G$
defined as the cardinality of a smallest set of edges $F\subseteq
E(\overline{G})$ such that $\gamma_{rR}(G+F)<\gamma_{rR}(G),$ where
$\overline{G}$ denotes the complement graph of $G.$ If there is
no subset of edges $F$ {satisfying }$\gamma_{rR}(G+F)<\gamma
_{rI}(G)$, {then }we define $r_{rR}(G)=0$. Since for any nontrivial
connected graph $G,$ $\gamma_{rR}(G)\geq2,$ we deduce that $r_{rR}(G)=0$ for
all nontrivial connected graphs with {$\gamma_{rR}(G)=2$}. Moreover, a subset
$E^{\prime}\subseteq E(\overline{G})$ {is called }an{ }$r_{rR}%
(G)$-set if $|E^{\prime}|=r_{rR}(G)$ and $\gamma_{rR}(G+E^{\prime}%
)<\gamma_{rR}(G)$.

Further, we will prove that the
decision problem  associated with the Restrained Roman reinforcement is NP-hard. Then various properties of the
restrained Roman reinforcement number are investigated and some sharp bounds
on it are presented.

We finish this section by observing that any $r_{rR}(G)$-set of a connected
graph $G$ with $\gamma_{rR}(G)\geq3$ can decrease the restrained Roman
domination number of $G$ by {at most two. }

\begin{prpen}
\label{frame}Let $G$ be a connected graph with $\gamma_{rR}(G)\geq3$. If $F$
is {an} $r_{rR}(G)$-set, then
\[
\gamma_{rR}(G)-2\leq\gamma_{rR}(G+F)\leq\gamma_{rR}(G)-1.
\]
{Both }bounds are sharp.
\end{prpen}
\proofen By assumption, $\gamma_{rR}(G+F)<\gamma_{rR}(G),$
whence the upper bound follows. To show the lower bound, let us assume that
$$\gamma_{rR}(G+F)\leq\gamma_{rR}(G)-3.$$ Let $f$ be a $\gamma_{rR}%
(G+F)$-function and let $uv\in F$ such that $0\in\{f(u),f(v)\}$. If such an
edge does not exist, then $f$ is an RRD-function of~$G$ leading to the
contradiction $$\gamma_{rR}(G)\leq\gamma_{rR}(G+F).$$
Hence we suppose that~$uv$ exists, and let~$F^{\prime}=F-\{uv\}.$ Without loss of
generality, {suppose} that $f(u)=0$. If $f(v)=1$, then $f$ is an RRD-function
of~$G$ leading to the contradiction $$\gamma_{rR}(G)\leq\gamma_{rR}(G+F)$$
too. {Hence }assume that $f(v)\neq1$.

First let $f(v)=2$. If $u$ has a neighbor $w$ in $G+F^{\prime}$ with
$f(w)\geq1$, then the function $g$ defined by $g(w)=2$ and $g(x)=f(x)$
otherwise, is an RRD-function of $G+F^{\prime}$ yielding {as above to the
contradiction} $\gamma_{rR}(G+F^{\prime})<\gamma_{rR}(G)$. Hence we assume
that each neighbor $u$ in $G+F^{\prime}$ is {assigned }0 under $f$. Let
$x_{1},\dots,x_{k}$ be the neighbors of $u$ in $G+F^{\prime}$. If $k=1$ and
$x_{1}$ has a neighbor assigned~0 other than $u$, then the function $g(u)=1$
and $g(x)=f(x)$ otherwise, is {an} RRD-function of $G+F^{\prime}$ yielding
$$\gamma_{rR}(G+F^{\prime})\leq\gamma_{rR}(G+F)+1<\gamma_{rR}(G),$$  this is a
contradiction. If $k=1$ and $x_{1}$ has no neighbor assigned~0 other than
$u$, then the function $g(u)=g(x_{1})=1$ and $g(x)=f(x)$ otherwise, is an
RRD-function of $G+F^{\prime}$ and thus $$\gamma_{rR}(G+F^{\prime})\leq
\gamma_{rR}(G+F)+2<\gamma_{rR}(G),$$ it is  a contradiction too. Hence assume that
$k\geq2$. If some~$x_{i}$ has no neighbor assigned~0 other than~$u,$ then
the function $g(x_{i})=2$ and $g(x)=f(x)$ otherwise, is an RRD-function of
$G+F^{\prime}$ yielding {again }$\gamma_{rR}(G+F^{\prime})<\gamma_{rR}(G)$.
Hence we assume that for each $i,$ $x_{i}$ has at least two neighbors
assigned $0$ under~$f.$ In this case,    we have $g(u)=1$ and $g(x)=f(x)$
otherwise, it is an RRD-function of $G+F^{\prime}$ and thus $$\gamma
_{rR}(G+F^{\prime})<\gamma_{rR}(G).$$

Finally, assume that $f(v)=0$. Since $F$ is an $r_{rR}(G)$-set, we {can
}suppose, without loss of generality, that all neighbors of $u$ in
$G+F^{\prime}$ have positive labels under $f$. Now, if $v$ has a neighbor
with weight~$0$ in $G+F^{\prime}$, then the function $g(u)=1$ and $g(x)=f(x)$
otherwise, it is an RRD-function of $G+F^{\prime}$ {while if }$v$ has no
neighbor with weight~$0$ in $G+F^{\prime}$, then the function $g(u)=g(v)=1$
and $g(x)=f(x)$ otherwise, is {an }RRD-function of $G+F^{\prime}.$ {Both
situations} yield the contradiction $\gamma_{rR}(G+F^{\prime})<\gamma
_{rR}(G).$ Consequently, $$\gamma_{rR}(G+F)\geq\gamma_{rR}(G)-2.$$

The upper bound of Proposition~\ref{frame} is
attained for the cycle $C_{4},$ while the lower bound is attained
for the cycle $C_{6}.$ \hfill$\square$\\[1ex]

%\begin{figure}[t]
%\centerline{
%\includegraphics[width=0.45\textwidth]{Fig1}}
%\caption{The graphs $L_{i}$ and $H=H_{1}\cup H_{2}$}
%\label{fig1}
%\end{figure}

\begin{figure}[h!]
\centering
\begin{tikzpicture}%[scale=.3, transform shape]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%(1)
\node [draw, shape=circle,fill=white,scale=0.3] (p1) at (-2.0,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (q1) at (-1.3,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (a1) at (-1.3,2.4) {};
\node [draw, shape=circle,fill=white,scale=0.3] (b1) at (-1.3,3) {};
%\node [draw, shape=circle,fill=white,scale=0.3] (d1) at (-1.3,3.7) {};
\node [draw, shape=circle,fill=black,scale=0.3] (r1) at (-.6,1.8) {};
\node [draw, shape=circle,fill=black,scale=0.3] (u1) at (-2.0,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (w1) at (-1.3,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (u'1) at (-.6,0) {};
\draw [line width=0.8pt](p1)--(q1)--(r1)--(u'1)--(w1)--(u1)--(p1)(w1)--(q1);
\draw [line width=0.8pt](b1)--(p1)--(a1)--(r1)--(b1);
\draw [line width=0.8pt](a1)--(r1);
\node at (-2.2,1.9) {$z_i$};
%\node at (-1.3,2.0) {$q_i$};
%\node at (-1.3,2.6) {$a_i$};
%\node at (-1.3,3.23) {$b_i$};
%\node at (-1.3,3.93) {$d_i$};
\node at (-.4,1.9) {$y_i$};
\node at (-2.0,-0.3) {$x_i$};
%\node at (-1.3,-0.3) {$w_i$};
\node at (-.5,-0.3) {$\overline{x}_i$};
\node at (-1.3,-1) {$L_i$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%(2)
\node [draw, shape=circle,fill=white,scale=0.3] (s1) at (3.2,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (s2) at (3.2,0.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (s3) at (3.2,1.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (s4) at (4.2,2){};
\node [draw, shape=circle,fill=white,scale=0.3] (t1) at (2.1,1){};
\node [draw, shape=circle,fill=white,scale=0.3] (t2) at (1.25,0.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t3) at (2.1,0){};
\node [draw, shape=circle,fill=white,scale=0.3] (p1) at (4.1,1){};
\node [draw, shape=circle,fill=white,scale=0.3] (p2) at (5.25,0.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p3) at (4.1,0){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v1) at (3.2,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v2) at (4.1,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v3) at (4.1,-6.4){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v4) at (3.2,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v5) at (2.3,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v6) at (2.3,-5.6){};
%\draw  [line width=0.8pt](c2)--(s1)--(c1)(s1)--(c3);
\draw  [line width=0.8pt](s2)--(t1)--(t2)--(t3)--(s2)--(t2);
\draw  [line width=0.8pt](s1)--(s2)--(s3);
\draw  [line width=0.8pt](s1)--(s3)--(s4)--(s1);
\draw  [line width=0.8pt](s2)--(p1)--(p2)--(p3)--(s2)--(p2);
\node at (2.95,2.8) {$s_1$};
\node at (2.9,.9) {$s_3$};
\node at (2.95,1.5) {$s_2$};
\node at (4.5,2) {$s_4$};
%\node at (2.95,-5.4) {$v_1$};
%\node at (4.35,-5.4) {$v_2$};
%\node at (4.35,-6.4) {$v_3$};
%\node at (3.2,-6.6) {$v_4$};
%\node at (2.05,-6.4) {$v_5$};
%\node at (2.05,-5.6) {$v_6$};
%\node at  (2.1,1.25) {$t_1$};
%\node at (1,.5) {$t_2$};
%\node at (2.1,-.3) {$t_3$};
%\node at  (4.1,1.25) {$m_1$};
%\node at (5.6,.5) {$m_2$};
%\node at (4.1,-.3) {$m_3$};
\node at (3.2,-1) {$H_1$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node [draw, shape=circle,fill=white,scale=0.3] (s11) at (9.2,2.7){};
\node [draw, shape=circle,fill=black,scale=0.3] (s21) at (9.2,.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (s31) at (9.5,1.5){};
%\node [draw, shape=circle,fill=white,scale=0.3] (s41) at (6.2,-3.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t11) at (8.1,1){};
\node [draw, shape=circle,fill=white,scale=0.3] (t21) at (7.25,.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t31) at (8.1,0){};
\node [draw, shape=circle,fill=white,scale=0.3] (p11) at (10,1){};
\node [draw, shape=circle,fill=white,scale=0.3] (p21) at (11,.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p31) at (10,0){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v1) at (3.2,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v2) at (4.1,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v3) at (4.1,-6.4){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v4) at (3.2,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v5) at (2.3,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v6) at (2.3,-5.6){};
%\draw  [line width=0.8pt](c2)--(s11)--(c1)--(s11)--(c3);
\draw  [line width=0.8pt](s21)--(t11)--(t21)--(t31)--(s21)--(t21);
\draw  [line width=0.8pt](s21)--(s11)--(s31)--(s21);
\draw  [line width=0.8pt](s21)--(p11)--(p21)--(p31)--(s21)--(p21);
\node at (8.9,2.8) {$s_1'$};
\node at (9,.9) {$s_3'$};
\node at (9.7,1.7) {$s_2'$};
%\node at (6.5,-3.5) {$s_4$};
%\node at (2.95,-5.4) {$v_1$};
%\node at (4.35,-5.4) {$v_2$};
%\node at (4.35,-6.4) {$v_3$};
%\node at (3.2,-6.6) {$v_4$};
%\node at (2.05,-6.4) {$v_5$};
%\node at (2.05,-5.6) {$v_6$};
%\node at  (8.1,1.25) {$t_1'$};
%\node at (7.1,.5) {$t_2'$};
%\node at (8.1,-.25) {$t_3'$};
%\node at  (10.1,1.25) {$m_1'$};
%\node at (11.4,.5) {$m_2'$};
%\node at (10.1,-.3) {$m_3'$};
\node at (9.2,-1) {$H_2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\caption{The graphs $L_{i}$ and $H=H_{1}\cup H_{2}$.}%
\label{FAZ}%
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{NP-hardness result}

The aim of this section, is to show that the decision problem associated with
the Restrained Roman reinforcement is NP-hard. Consider the following decision
problem.\\
\medskip

\textbf{Restrained Roman reinforcement problem (RR-reinforcement)}
\medskip

\textbf{Instance:} \emph{A nonempty graph $G$ and a positive integer $k$.}

\textbf{Question:} \emph{Is }$r_{rR}(G)\leq k$?

\medskip

We show that the NP-hardness of the RR-reinforcement problem by transforming
the well-known 3-SAT problem to it in polynomial time. Recall that the 3-SAT
problem specified below was proven to be NP-complete in \cite{GJ}.\\

\medskip
\textbf{3-SAT problem}

\medskip
%\subsection{3-SAT problem}
\textbf{Instance: } A collection $\mathcal{C}=\{C_{1}%
,C_{2},\ldots,C_{m}\}$ of clauses over a finite set $X$ of variables such that
$|C_{j}|=3$ for every $j\in\{1,2,\ldots,m\}.$

\textbf{Question: } Is there a truth assignment for $X$
that satisfies all the clauses in $\mathcal{C}$?

\begin{theoen}
Problem {RR-reinforcement }is NP-hard for an arbitrary graph.
\end{theoen}
\proofen
Let $X=\{x_{1},x_{2},\ldots,x_{n}\}$ and
$\mathcal{C}=\{C_{1},C_{2},\ldots,C_{m}\}$ be an arbitrary instance of 3-SAT
problem. We will build a graph $G$ and a positive integer $k$ such that
$r_{rR}(G)\leq k$ if and only if $\mathcal{C}$ is satisfiable.

For each $i\in\{1,2,\ldots,n\},$ we associate to
the variable~$x_{i}\in X$ a copy of the graph~$L_{i}$
as depicted in Figure~\ref{FAZ}, and for each $j\in
\{1,2,\ldots,m\}$, we associate to the clause $C_{j}%
=\{u_{j},v_{j},w_{j}\}\in\mathcal{C}$ a vertex~$c_{j}$ by adding the
edge-set $E_{j}=\{c_{j}u_{j},c_{j}v_{j},c_{j}w_{j}\}$. Finally, we enclose
the graph $H$ illustrated in Figure~\ref{FAZ} by connecting vertices
$s_{1},s_{1}^{\prime}$ to every vertex $c_{j}$. Clearly, the resulting graph
$G$ is of order $8n+m+19$ and size~$11n+5m+27$ and hence~$G$ can be
built in polynomial time. Set $k=1$. Figure~\ref{FAZ1} provides
an example of the resulting graph when $X=\{x_{1},x_{2},x_{3},x_{4}\}$ and
$\mathcal{C}=\{C_{1},C_{2},C_{3}\}$, where $C_{1}=\{x_{1},x_{2},\bar{x}_{3}%
\}$, $C_{2}=\{\bar{x}_{1},x_{2},x_{4}\}$ and $C_{3}=\{\bar{x}_{2}%
,x_{3},x_{4}\}$.

\begin{figure}[h]
\centering
\begin{tikzpicture}%[scale=.3, transform shape]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%(1)
\node [draw, shape=circle,fill=white,scale=0.3] (p1) at (-2.0,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (q1) at (-1.3,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (a1) at (-1.3,2.4) {};
\node [draw, shape=circle,fill=white,scale=0.3] (b1) at (-1.3,3) {};
\node [draw, shape=circle,fill=black,scale=0.3] (r1) at (-.6,1.8) {};
\node [draw, shape=circle,fill=black,scale=0.3] (u1) at (-2.0,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (w1) at (-1.3,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (u'1) at (-.6,0) {};
\draw [line width=0.8pt](p1)--(q1)--(r1)--(u'1)--(w1)--(u1)--(p1)(w1)--(q1);
\draw [line width=0.8pt](b1)--(p1)--(a1)--(r1)--(b1);
\node at (-2.2,1.9) {$z_1$};
%\node at (-1.3,2.0) {$q_1$};
%\node at (-1.3,2.6) {$a_1$};
%\node at (-1.3,3.23) {$b_1$};
\node at (-.4,1.9) {$y_1$};
\node at (-2.0,-0.3) {$x_1$};
%\node at (-1.3,-0.3) {$w_1$};
\node at (-.5,-0.3) {$\overline{x}_1$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%(2)
\node [draw, shape=circle,fill=black,scale=0.3] (p2) at (0.9,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (q2) at (1.6,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (a2) at (1.6,2.4) {};
\node [draw, shape=circle,fill=white,scale=0.3] (b2) at (1.6,3) {};
\node [draw, shape=circle,fill=white,scale=0.3] (r2) at (2.3,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (u2) at (0.9,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (w2) at (1.6,0) {};
\node [draw, shape=circle,fill=black,scale=0.3] (u'2) at (2.3,0) {};
\draw [line width=0.8pt](p2)--(q2)--(r2)--(u'2)--(w2)--(u2)--(p2);
\draw [line width=0.8pt](b2)--(p2)--(a2)--(r2)--(b2)(w2)--(q2);
\node at (0.7,2) {$z_2$};
%\node at (1.6,2) {$q_2$};
%\node at (1.6,2.6) {$a_2$};
%\node at (1.6,3.23) {$b_2$};
\node at (2.5,2) {$y_2$};
\node at (0.7,-0.25) {$x_2$};
%\node at (1.6,-0.250) {$w_2$};
\node at (2.3,-0.25) {$\overline{x}_2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%(3)
\node [draw, shape=circle,fill=white,scale=0.3] (p3) at (3.8,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (q3) at (4.5,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (a3) at (4.5,2.4) {};
\node [draw, shape=circle,fill=white,scale=0.3] (b3) at (4.5,3) {};
\node [draw, shape=circle,fill=black,scale=0.3] (r3) at (5.2,1.8) {};
\node [draw, shape=circle,fill=black,scale=0.3] (u3) at (3.8,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (w3) at (4.5,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (u'3) at (5.2,0){};
\draw  [line width=0.8pt](u3)--(p3)--(q3)--(r3)--(u'3)--(w3)--(u3);
\draw [line width=0.8pt](b3)--(p3)--(a3)--(r3)--(b3)(w3)--(q3);
\node at(3.6,2)  {$z_3$};
%\node at (4.5,2) {$q_3$};
%\node at (4.5,2.6) {$a_3$};
%\node at (4.5,3.23) {$b_3$};
\node at (5.4,2) {$y_3$};
\node at (3.8,-0.25) {$x_3$};
%\node at (4.5,-0.25)  {$w_3$};
\node at (5.2,-0.25) {$\overline{x}_3$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%(4)
\node [draw, shape=circle,fill=white,scale=0.3] (p4) at (6.7,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (q4) at (7.4,1.8) {};
\node [draw, shape=circle,fill=white,scale=0.3] (a4) at (7.4,2.4) {};
\node [draw, shape=circle,fill=white,scale=0.3] (b4) at (7.4,3) {};
\node [draw, shape=circle,fill=black,scale=0.3] (r4) at (8.1,1.8) {};
\node [draw, shape=circle,fill=black,scale=0.3] (u4) at (6.7,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (w4) at (7.4,0) {};
\node [draw, shape=circle,fill=white,scale=0.3] (u'4) at (8.1,0) {};
\draw  [line width=0.8pt](u4)--(p4)--(q4)--(r4)--(u'4)--(w4)--(u4);
\draw [line width=0.8pt](b4)--(p4)--(a4)--(r4)--(b4)(w4)--(q4);
\node at(6.5,2) {$z_4$};
%\node at (7.4,2) {$q_4$};
%\node at (7.4,2.6) {$a_4$};
%\node at (7.4,3.23) {$b_4$};
\node at (8.3,2) {$y_4$};
\node at (6.8,-0.2) {$x_4$};
%\node at (7.4,-0.2) {$w_4$};
\node at (8.1,-0.2) {$\overline{x}_4$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node [draw, shape=circle,fill=white,scale=0.3] (c1) at (0.8,-2.2){};
\node [draw, shape=circle,fill=white,scale=0.3] (c2) at (3.2,-2.2){};
\node [draw, shape=circle,fill=white,scale=0.3] (c3) at(5.6,-2.2){};
\node at (0.6,-1.8) {$c_1$};
\node at (3.2,-1.9) {$c_2$};
\node at (5.55,-1.85) {$c_3$};
\draw  [line width=0.8pt](u2)--(c1)--(u1)(c1)--(u'3);
\draw  [line width=0.8pt](u'1)--(c2)--(u2)(c2)--(u4);
\draw  [line width=0.8pt](u'2)--(c3)--(u3)(c3)--(u4);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node [draw, shape=circle,fill=black,scale=0.3] (s1) at (0.2,-3.2){};
\node [draw, shape=circle,fill=black,scale=0.3] (s2) at (0.2,-5){};
\node [draw, shape=circle,fill=white,scale=0.3] (s3) at (0.2,-4){};
\node [draw, shape=circle,fill=white,scale=0.3] (s4) at (1.2,-3.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t1) at (-1.1,-4.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t2) at (-2.25,-5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t3) at (-1.1,-5.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p1) at (1.1,-4.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p2) at (2.25,-5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p3) at (1.1,-5.5){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v1) at (3.2,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v2) at (4.1,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v3) at (4.1,-6.4){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v4) at (3.2,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v5) at (2.3,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v6) at (2.3,-5.6){};
\draw  [line width=0.8pt](c2)--(s1)--(c1)(s1)--(c3);
\draw  [line width=0.8pt](s2)--(t1)--(t2)--(t3)--(s2)--(t2);
\draw  [line width=0.8pt](s1)--(s2)--(s3);
\draw  [line width=0.8pt](s1)--(s3)--(s4)--(s1);
\draw  [line width=0.8pt](s2)--(p1)--(p2)--(p3)--(s2)--(p2);
\node at (0,-3.3) {$s_1$};
\node at (0,-4.7) {$s_3$};
\node at (0,-4) {$s_2$};
\node at (1.5,-3.5) {$s_4$};
%\node at (2.95,-5.4) {$v_1$};
%\node at (4.35,-5.4) {$v_2$};
%\node at (4.35,-6.4) {$v_3$};
%\node at (3.2,-6.6) {$v_4$};
%\node at (2.05,-6.4) {$v_5$};
%\node at (2.05,-5.6) {$v_6$};
%\node at  (-1.1,-4.25) {$t_1$};
%\node at (-2.4,-5) {$t_2$};
%\node at (-1.1,-5.7) {$t_3$};
%\node at  (1.1,-4.25) {$m_1$};
%\node at (2.6,-5) {$m_2$};
%\node at (1.1,-5.7) {$m_3$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node [draw, shape=circle,fill=white,scale=0.3] (s11) at (6.2,-3.2){};
\node [draw, shape=circle,fill=black,scale=0.3] (s21) at (6.2,-5){};
\node [draw, shape=circle,fill=white,scale=0.3] (s31) at (6.5,-4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (s41) at (6.2,-3.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t11) at (5.1,-4.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t21) at (4.25,-5){};
\node [draw, shape=circle,fill=white,scale=0.3] (t31) at (5.1,-5.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p11) at (7,-4.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p21) at (8,-5){};
\node [draw, shape=circle,fill=white,scale=0.3] (p31) at (7,-5.5){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v1) at (3.2,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v2) at (4.1,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v3) at (4.1,-6.4){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v4) at (3.2,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v5) at (2.3,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v6) at (2.3,-5.6){};
\draw  [line width=0.8pt](c2)--(s11)--(c1)--(s11)--(c3);
\draw  [line width=0.8pt](s21)--(t11)--(t21)--(t31)--(s21)--(t21);
\draw  [line width=0.8pt](s21)--(s11)--(s31)--(s21);
\draw  [line width=0.8pt](s21)--(p11)--(p21)--(p31)--(s21)--(p21);
\node at (6.6,-3.3) {$s_1'$};
\node at (6,-4.7) {$s_3'$};
%\node at (6.8,-3.8) {$s_2'$};
%\node at (6.5,-3.5) {$s_4$};
%\node at (2.95,-5.4) {$v_1$};
%\node at (4.35,-5.4) {$v_2$};
%\node at (4.35,-6.4) {$v_3$};
%\node at (3.2,-6.6) {$v_4$};
%\node at (2.05,-6.4) {$v_5$};
%\node at (2.05,-5.6) {$v_6$};
%\node at  (5.5,-4.25) {$t_1'$};
%\node at (4.1,-5) {$t_2'$};
%\node at (5.1,-5.8) {$t_3'$};
%\node at  (7.1,-4.25) {$m_1'$};
%\node at (8.4,-5) {$m_2'$};
%\node at (7.1,-5.7) {$m_3'$};
\end{tikzpicture}
\caption{An instance of the restrained Roman reinforcement number problem
resulting from an instance of 3-SAT. Here $k=1$ and $\gamma_{rR}(G)=22$, where
the black vertex $p$ means there is a RRDF $f$ with $f(p)=2$.}%
\label{FAZ1}%
\end{figure}
It is easy to verify that for any $\gamma_{rR}(G)$-function $g$ we
must have $$\sum_{v\in V(L_{j})}g(v)\geq4$$ for each $j\in\{1,2,\ldots,n\}$.
Moreover, to restrained Roman dominate all vertices of $V(H)$, we need
that $$\sum_{i=1}^{m}g(c_{i})+g(V(H))\geq6.$$
Therefore $$\gamma_{rR}(G)=w(g)\geq4n+6.$$ Basing on
the assignment given to the graph in Figure~\ref{FAZ1}, one can easily define
an RRD-function of $G$ with weight $4n+6$, which
consequently leads to $\gamma_{rR}(G)=4n+6$.

In the following, we  show that $\mathcal{C}$ is satisfiable if and only if
$r_{rR}(G)=1$. Let $\mathcal{C}$ be satisfiable and $t:X\rightarrow\{T,F\}$
a satisfying function for $\mathcal{C}$. We build a subset $S$ of vertices of
$G$ as follows. If $t(x_{i})=T$, then put the vertices $x_{i}$ and $y_{i}$ in
$S$; while if $t(x_{i})=F$, then put the vertices $\overline{x_{i}}$ and
$z_{i}$ in $S$. So $|S|=2n$. Define the function $h$ on $V(G)$ by $h(x)=2$
for every $x\in S,$ $h(s_{1})=1,$ $h(s_{3})=h(s_{3}^{\prime})=2$ and $h(y)=0$ for
the remaining vertices. It is easy to verify that $h$ is an RRD-function of
$G+s_{4}s_{3}$ of weight $$4n+5<\gamma_{rR}(G)=4n+6,$$ and hence
$r_{rR}(G)=1$.

Conversely, let $r_{rR}(G)=1$. Then, there is an edge $e=uv\in E(\overline
{G})$ for which $$\gamma_{rR}(G+e)<4n+6.$$ Let $g=(V_{0},V_{1},V_{2})$ be a
$\gamma_{rR}(G+e)$-function. Since whatever the added edge $e,$ we have
${g(V(L_{i}))\geq4}$, {and thus vertices }$u$ and $v$ cannot both belong to
$V(L_{i})$ (for otherwise ${\gamma_{rR}(G+e)\geq4n+6}$). On the
other hand, since $r_{rR}(G)=1$ {and }$g(V(L_{i}))\geq4$, we must
have $$\sum_{j=1}^{m}g(c_{j})+g(V(H))<6.$$ Since also whatever the
added edge $e,$ we have $g(V(H))\geq5,$ we conclude that $g(V(H))=5.$
In particular, this is only possible if $g(s_{1}^{\prime})=0,$
$g(s_{1})\leq1$ {and} $$\sum_{j=1}^{m}g(c_{j})=0.$$ {In addition, we note that
}if $\{x_{i},\bar{x_{i}}\}\subseteq V_{2}$ or $\{x_{i},\bar{x_{i}}\}\cap
V_{1}\neq\emptyset$ for some $i$, then $g(V(L_{i}))\geq5$ which results in the
contradiction $$\gamma_{rR}(G+e)\geq4n+6.$$ Thus, $\left\vert \{x_{i}%
,\bar{x_{i}}\}\cap V_{2}\right\vert \leq1$ {and }$\{x_{i},\bar{x_{i}}\}\cap
V_{1}=\emptyset$ for {every }$i\in\{1,\dots,n\}$. Therefore {each} vertex
$c_{j}$ {must have a neighbor in }$\{x_{i},\bar{x_{i}}\}$ for some $i$ {which
is assigned a 2.} {In this case, define the mapping }%
$t:X\rightarrow\{T,F\}$ by
\begin{equation}
t(x_{i})=%
\begin{cases}
T & \text{if \ \ $f(x_{i})=2$,}\\
F & \text{otherwise \ \ }%
\end{cases}
\label{eq2}%
\end{equation}
{for $i\in\{1,\ldots,n\}$}.

% We shall show that $t$ is a satisfying truth
%assignment for $\mathcal{C}$. It is enough to show that every clause in
%$\mathcal{C}$ is satisfied by $t$. Consider arbitrary clause
%$C_{j}\in\mathcal{C}$ {for some }$j\in\{1,\dots,m\}$. If $c_{j}$ is dominated by
%$x_{i}$, then $g(x_{i})=2$ and so $t(x_{i})=T$.
%If $c_{j}$ is dominated by
%$\bar{x_{i}}$, then $g(\bar{x_{i}})=2$ and hence $t(x_{i})=F$ and
%$t(\bar{x_{i}})=T$. Therefore, in {either case} the clause $C_{j}$ is
%satisfied. The arbitrariness of $j$ shows that all clauses in $\mathcal{C}$
%are satisfied by $t$, that is, $\mathcal{C}$ is satisfiable. {This completes
%the proof of the }theorem. \hfill$\square$\\[1ex]
We show that t satisfies the truth assignment for $\mathcal{C}$.
 It is enough to show that every clause in $\mathcal{C}$ is satisfied by~$t$.
 Consider an arbitrary clause $C_{j}\in\mathcal{C}$ for some $j\in\{1,\dots,m\}$.
If $c_{j}$ is dominated by~$x_{i}$, then $g(x_{i})=2$ and so $t(x_{i})=T$.
If $c_{j}$ is dominated by $\bar{x_{i}}$, then $g(\bar{x_{i}})=2$ and hence $t(x_{i})=F$  and $t(\bar{x_{i}})=T$. Therefore, in either case the clause $C_{j}$  is satisfied. The arbitrariness of $j$ shows that all clauses in $\mathcal{C}$ are satisfied by $t$, that is, $\mathcal{C}$ is satisfiable. This completes the proof of the theorem. \hfill$\square$\\[1ex]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Exact values}

In this section, we determine the restrained Roman reinforcement number of
some classes of graphs including paths, cycles and complete $p$-partite graphs
for any integer~$p\geq2$.
%{Here, we recall some results that we will use later.
As observed in~\cite{PP}, for every connected graph $G$ of order $n\geq2,$ we have
$2\leq\gamma_{rR}(G)\leq n.$ {A characterization of all connected graphs of
order }$n$ {with }$\gamma_{rR}(G)\in\{2,3,n\}$ was provided  in
\cite{PP,AAC2} as follows.

{{ \begin{figure}[h]
\centering
\begin{tikzpicture}%[scale=.3, transform shape]
\node [draw, shape=circle,fill=black,scale=0.5] (a1) at  (1,0) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a2) at  (3,0) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a3) at  (3,1) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a4) at  (2,1.5) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a5) at  (1,1) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a6) at  (0,1.6) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a7) at  (0,1.2) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a8) at  (0,0.8) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a9) at  (0,0.4) {};
\draw(a9)--(a5)--(a8)--(a5)--(a7)--(a5)--(a6)--(a5)--(a4)--(a3)--(a2)--(a1)--(a5);
\node at (2,-.9) {The graph $B_4$.};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node [draw, shape=circle,fill=black,scale=0.5] (a1) at  (5,0) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a2) at  (7,0) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a3) at  (7,1) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a4) at  (6,1.5) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a5) at  (5,1) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a6) at  (4,1.4) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a7) at  (4,1) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a8) at  (4,.6) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a9) at  (8,1.2) {};
\node [draw, shape=circle,fill=black,scale=0.5] (a10) at  (8,.8) {};
\draw(a8)--(a5)--(a7)--(a5)--(a6)--(a5)--(a4)--(a3)--(a10)--(a3)--(a9)--(a3)--(a2)--(a1)--(a5);
\node at (6,-.9) {The graph $B_{3,2}$.};
\node at (1,1.3) {$u_1$};
\node at (5,1.3) {$u_1$};
\node at (2,1.8) {$u_2$};
\node at (6,1.8) {$u_2$};
\node at (7,1.3) {$u_3$};
\node at (3,1.3) {$u_3$};
\node at (1,-.3) {$u_5$};
\node at (5,-.3) {$u_5$};
\node at (7,-.3) {$u_4$};
\node at (3,-.3) {$u_4$};
\end{tikzpicture}
\caption{Graphs $B_{4}$ and $B_{3,2}$.}%
\label{fig-0}%
\end{figure}} }

Let $C:=(u_{1}u_{2}u_{3}u_{4}u_{5})$ be a cycle of length 5 and let $B_{p}$ be
the graph obtained from $C$ by adding $p\geq1$ new
vertices attached by edges at $u_{1}$ and let $B_{p,q}$ be
the graph obtained from $C$ by adding $p\geq1$ new
vertices attached by edges at $u_{1}$ and $q\geq1$ other new
vertices attached by edges at $u_{3}$ (see Fig.~\ref{fig-0}).
Recall that the
\textit{diameter,} \textrm{diam}$(G),$ of a graph $G$
is the maximum distance between the pair of vertices.

\begin{prpen}
[\cite{PP}]\label{n}Let $G$ be a connected graph of order $n\geq2$. Then
\begin{enumerate}
  \item[\textup{(a)}] $\gamma_{rR}(G)=2$ if and only if $n=2$ or $\Delta(G)=n-1$ and
$\delta(G)\geq2$\textup;
  \item[\textup{(b)}] $\gamma_{rR}(G)=n$ if and only if $G\simeq C_{4},C_{5},B_{p},B_{p,q}$ or $G$ is a tree with $\mathrm{diam}(G)\leq5$.
\end{enumerate}
\end{prpen}

\begin{prpen}
[\cite{AAC2}]\label{pp1} Let $G$ be a connected graph of order $n\geq4$.
Then $\gamma_{rR}(G)=3$ if and only if $G$ satisfies one of the following
conditions\textup:
\begin{itemize}
\item[\emph{(i)}] $\Delta(G)=n-1$ and $G$ has exactly one leaf\textup;
\item[\emph{(ii)}] $\Delta(G)=n-2$ and $G$ has a vertex $u$ of degree $n-2$
such that the induced subgraph $G[N(u)]$ has no isolated vertex.
\end{itemize}
\end{prpen}

On the other hand, the exact values of the restrained Roman domination
number have been established in \cite{PP} for paths, cycles and complete
$p$-partite graphs.

\begin{prpen}[\cite{PP}]\label{o3}   The following conditions holds\textup:
\begin{enumerate}
\item[\emph{(a)}] $\gamma_{rR}(P_{n})=n$\ for\ $1\leq n\leq6$\ and \
$\gamma_{rR}(P_{n})=\left\lceil {(2n+1)}/{3}\right\rceil +1$ for $n\geq 7;$
\item[\emph{(b)}]$\gamma_{rR}(C_{n})=2\left\lceil {n}/{3}\right\rceil $\ when\
$n\not \equiv 2\;(\mathrm{mod}\;3)$\ and\ $\gamma_{rR}(C_{n})=2\left\lceil
{n}/{3}\right\rceil +1$ otherwise\textup;
\item[\emph{(c)}] $\gamma_{rR}(K_{m,n})=4$\ for\
$m,n\geq2;$
\item[\emph{(d)}] if $K_{n_{1},n_{2},\ldots,n_{p}}$ {is }the complete
$p$-partite graph such that $p\geq3$ and $n_{1}\leq n_{2}\leq\ldots\leq n_{p},$ then \ $\gamma_{rR}(K_{1,n_{2},\ldots,n_{p}})=2,\ $ $\gamma_{rR}%
(K_{2,n_{2},\ldots,n_{p}})=3$\ and\  $\gamma_{rR}(K_{n_{1},n_{2},\ldots,n_{p}})=4$ for $n_{1}\geq3$.
\end{enumerate}
\end{prpen}

Now we are ready to find the restrained Roman reinforcement number for
paths, cycles and complete $p$-partite graphs, $p\ge 2$.

\begin{prpen}
\label{pn1} For $n\geq3$,\quad $r_{rR}$\emph{$(P_{n})=1$.}
\end{prpen}
\proofen Let $P_{n}:=w_{1}w_{2}\ldots w_{n}$. If
$n\equiv0\,(\mathrm{{mod}\,3})$, then the function $g$ defined by
$$g(w_{3i+1})=2$$ for $0\leq i\leq(n-3)/3$ and $g(w)=0$ otherwise, is an
RRD-function of $P_{n}+w_{1}w_{n}$ of weight ${2n}/{3}.$ If~$n\equiv
2\,(\mathrm{{mod}\ 3)}$, then the function~$g$ defined by $g(w_{n})=2,$ $g(w_{3i+1})=2$ for $0\leq i\leq(n-5)/3$  and $g(x)=0$ otherwise, is {an}
RRD-function of $P_{n}+w_{1}w_{n-2}$ of weight $({2n+2})/{3}.$ Finally, if
$n\equiv1\,(\mathrm{{mod}\,3)}$, then the function $g$ defined by $g(w_{n}%
)=1$, $g(w_{3i+1})=2$ for $0\leq i\leq(n-4)/3$ and $g(w)=0$ otherwise, is {an}
RRD-function of $P_{n}+w_{1}w_{n-1}$ of weight $({2n+1})/{3}.$
All considered cases show that $r_{rR}(P_{n})=1.$ \hfill$\square$\


\begin{prpen}
\label{cn1}\emph{For $n\geq4$,
\[
r_{rR}(C_{n})=\left\{
\begin{array}
[c]{ll}%
2 & \text{\textrm{if }}n\equiv0\,(\mathrm{{mod}\,3}),\\
1 & \mathrm{otherwise.}%
\end{array}
\right.
\]
}
\end{prpen}
\proofen
 Assume that $C_{n}:=(w_{1}w_{2}\ldots w_{n})$ be a
cycle on $n$ vertices. If $n\equiv r\pmod 3$ with $r\in\{1,2\}$, then {by a
similar argument to that used in }the proof of Proposition \ref{pn1}, we can
see that $r_{rR}(C_{n})=1$. Hence we assume that $n\equiv0\pmod 3$. First,
{since }the function $g$ defined by $g(w_{n-2})=1$, $g(w_{3i+1})=2$ for $0\leq
i\leq(n-6)/3$ and $g(x)=0$ otherwise, is {\color{red}}an RRD-function of
$C_{n}+\{w_{1}w_{n-1},w_{1}w_{n-3}\}$ of weight $(2n-3)/3=\gamma_{rR}%
(C_{n})-1$ (Proposition \ref{o3}-(b)), {we deduce that }$r_{rR}(C_{n})\leq2$.

Now we prove the inverse inequality. For this purpose, we need
only to show that adding an arbitrary edge $e$ cannot decrease %
$\gamma_{rR}(C_{n}).$ {Observe that }for any edge $e\in\overline{C_{n}}$,
$$\gamma_{rR}(C_{n}+e)\leq\gamma_{rR}(C_{n}).$$ Let $e$ be an arbitrary edge in
$\overline{C_{n}}$ and let $f$ be a $\gamma_{rR}(C_{n}+e)$-function. {Suppose
first that there are }three consecutive vertices $w_{i},w_{i+1},w_{i+2}$ such
that $f(w_{i})=f(w_{i+1})=f(w_{i+2})=0$, say {for }$i=1$. Then the edge $e$
must join $w_{2}$ to {some }vertex {assigned 2, say }$w_{k}$, with %
$k\notin\{1,3\}.$ Also, to {restrained Roman }dominate $w_{1}$
and $w_{3}$, we must {also }have $f(w_{4})=f(w_{n})=2$.

Consider the cycles
$C^{\prime}:=(w_{2}w_{3}\ldots w_{k})$ {of order }$k-1$ {and }$C^{\prime
\prime}:=(w_{2}w_{k}\ldots w_{n}w_{1})$ {of order }$n-k+3.$ {Let }$k-1\equiv
s_{1}\pmod 3${ and }$n-k+3\equiv s_{2}\pmod 3.$ Notice that $s_{1}=0$ and
$s_{2}=2;$ $s_{1}=2$ and $s_{2}=0$ or $s_{1}=s_{2}=1.$ Assume that
$k-1\equiv0\pmod 3$ (the case $n-k+3\equiv0\pmod 3$ is similar). Then
$n-k+3\equiv2\pmod 3,$ and since the restrictions of $f$ on $V(C^{\prime})$
and $V(C^{\prime\prime})$ are RRD-functions, we deduce from Proposition
\ref{o3}-(b), that %
\begin{gather*}
\gamma_{rR}(C_{n}+e)    =f(V(C^{\prime}))+f(V(C^{\prime\prime}))-2\\
  \geq\gamma_{rR}(C^{\prime})+\gamma_{rR}(C^{\prime\prime})-2=\frac
{2(k-1)}{3}+\frac{2(n-k+3)+3+2}{3}-2
  =\frac{2n+3}{3}>\gamma_{rR}(C_{n}).{\ }%
\end{gather*}
Assume now that $s_{1}=s_{2}=1.$ Then, as above, it follows from Proposition~\ref{o3}-(b) that
\begin{gather*}
\gamma_{rR}(C_{n}+e)    =f(V(C^{\prime}))+f(V(C^{\prime\prime}))-2\\
  \geq\gamma_{rR}(C^{\prime})+\gamma_{rR}(C^{\prime\prime})-2=\frac
{2(k-1)+3+1}{3}+\frac{2(n-k+3)+3+1}{3}-2
  =\frac{2n+6}{3}>\gamma_{rR}(C_{n}).%
\end{gather*}
Thus in either case we obtain a contradiction.
%$\gamma_{rI}(C_{n}+e)=\gamma_{rI}(C_{n}).$
Next suppose there are three consecutive vertices $w_{i},w_{i+1},w_{i+2}$
such that $f(w_{i})+f(w_{i+1})+f(w_{i+2})=1$, say {for} $i=1$.

If $f(w_{2})=1$, then $f(w_{1})=f(w_{3})=0$ and each of $w_{1}$ and $w_{3}$ must be
adjacent a vertex assigned~$2$ as well as to a vertex assigned~$0$.
This
possible only if $e=w_{1}w_{3}$ and so $$H=(C_{n}+e)-w_{2}$$ is a cycle on
$n-1$ vertices, where the restriction of $f$ to $H$ is an RRD-function. It
follows that $$\gamma_{rR}(C_{n}+e)=f(V(H))+1\geq\gamma_{rR}(H)+1,$$ and by
Proposition~\ref{o3}-(b), we obtain $$\gamma_{rR}(C_{n}+e)\geq
\frac{2(n-1)+3+2}{3}+1>\gamma_{rR}(C_{n})$$ which is a contradiction. Hence
we can assume that $f(w_{2})=0$. Without loss of generality, let
$f(w_{1})=1$ and $f(w_{3})=0$. To restrained Roman dominate
$w_{2}$, the edge $e$ must join $w_{2}$ to a vertex with label~$2$, say $w_{k}$.
Likewise for $w_{3}$ we must have $f(w_{4})=2$. Now, consider the cycles
$C^{\prime}:=(w_{2}w_{3}\ldots w_{k})$ of order $k-1$ and the path
$P^{\prime}:=w_{k}\ldots w_{n}w_{1}$ of order $n-k+2.$

Let $k-1\equiv s_{1}\pmod 3$ and $n-k+2\equiv s_{2}\pmod 3.$  Notice that $s_{1}=0$ and
$s_{2}=1;$ $s_{1}=1$ and $s_{2}=0$ or $s_{1}=s_{2}=2.$ {Notice also that the
restrictions of }$f$ on $V(C^{\prime})$ and $V(P^{\prime})$ are
RRD-functions, and thus $$\gamma_{rR}(C_{n}+e)=f(V(C^{\prime}))+f(V(P^{\prime
}))-2\geq\gamma_{rR}(C^{\prime})+\gamma_{rR}(P^{\prime})-2.$$ Now using
Propositions \ref{o3}-(a,b), {we get a contradiction as before.

Finally, let
$$f(w_{i})+f(w_{i+1})+f(w_{i+2})\geq2$$ for each $1\leq i\leq n,$ where the sum
in indices is taken modulo $n$. Then we have
\[
\gamma_{rR}(C_{n}+e)=\frac{1}{3}\sum_{i=1}^{n}(f(w_{i})+f(w_{i+1}%
)+f(w_{i+2}))\geq\frac{2n}{3}=\gamma_{rR}(C_{n}),
\]
and therefore, $\gamma_{rR}(C_{n}+e)=\gamma_{rR}(G).$
%we deduce as before that for each
%situation }$\gamma_{rI}(C_{n}+e)\geq\gamma_{rI}(C_{n}).${ }
Consequently, $r_{rR}(C_{n})=2$ as desired. \hfill$\square$


\begin{prpen}
\label{completeb}\emph{For integers $1\leq r\leq s$ with $r+s\ge3$,
\[
r_{rR}(K_{r,s})=\left\{
\begin{array}
[c]{lll}%
1 & \mathrm{if} & r=1,2,3\hfill,\\
r-2 & \mathrm{if} & r\geq4.\hfill
\end{array}
\right.
\]
}
\end{prpen}
\proofen
Let $X=\{x_{1},x_{2},\ldots,x_{r}\}$ and
$Y=\{y_{1},y_{2},\ldots,y_{s}\}$ be the {partite sets }of $K_{r,s}$.

 If $r=1$,
then the function $g$ defined by $g(x_{1})=2$, $g(y_{1})=g(y_{2})=0$ and
$g(x)=1$ otherwise, is {an} RRD-function of $K_{1,s}+y_{1}y_{2}$ of weight
$n-1$ and it follows from Proposition \ref{n}-{(b)} that
$r_{rR}(K_{1,s})=1$.

 If $r=2$, then the function $g$ defined by $g(x_{1})=2$
and $g(x)=0$ otherwise, is {an} RRD-function of $K_{2,s}+x_{1}x_{2}$ of weight
$2$ and we get from Proposition \ref{n}-{(a)} that $r_{rR}%
(K_{2,s})=1$.

 If $r=3$, then the function $g$ defined by $g(x_{1}%
)=2,g(x_{2})=1$ and $g(x)=0$ otherwise, is {an }RRD-function of $K_{3,s}%
+x_{1}x_{3}$ of weight $3$ and by Proposition \ref{o3}-(c), we have
$r_{rR}(K_{3,s})=1$.

Let $r\geq4$. First we observe that the function $g$ defined by $g(x_{1})=2,$ $g(x_{2})=1$
and $g(x)=0$ otherwise, is an RRD-function of $K_{r,s}
+\{x_{1}x_{i}\mid3\leq i\leq r\}$ of weight $3$ and {thus }by Proposition~\ref{o3}-(c), $r_{rR}(K_{r,s})\leq r-2$.

To show that
$r_{rR}(K_{r,s})\geq r-2$, {let }$F$ be an $r_{rR}(K_{r,s})$-set.
Then $$2\leq\gamma_{rR}(K_{r,s}+F)\leq3.$$ By Propositions~\ref{n}%
-{(a) and \ref{pp1}} we must have $\Delta(K_{r,s}+F)\geq r+s-2$ and
this implies that $|F|\geq r-2$. Therefore $r_{rR}(K_{r,s})=r-2$
and the proof is complete. \hfill$\square$
%Hence assume that $\gamma_{rR}(K_{r,s}+F)=3$. {By
%Proposition \ref{33}, }$K_{p,q}+F\in\Psi\cup(\Theta\setminus\Omega)$. If
%$K_{p,q}+F\in\Psi$, then $\Delta(K_{p,q}+F)\geq p+q-2$ {and thus }$|F|\geq
%p-2$. {Hence suppose }that $K_{p,q}+F\in\Theta\setminus\Omega$ and let $x,y,z$
%be the vertices of $H$ (see the construction of graphs in Family $\Theta$). If
%$|X\cap\{x,y,z\}|=3$ (the case $|Y\cap\{x,y,z\}|=3$ is similar), then each
%vertex in $X-\{x,y,z\}$ must be adjacent to at least to two vertices in
%$\{x,y,z\}$ implying that $|F|\geq2(p-3)\geq p-2$. {Also, if }$|X\cap
%\{x,y,z\}|=2$ (the case $|Y\cap\{x,y,z\}|=2$ is similar), then each vertex in
%$X-\{x,y,z\}$ must be adjacent to at least to one vertex in $\{x,y,z\}$
%implying that $|F|\geq p-2$. {In any case, we conclude that }$r_{rI}%
%(K_{3,q})=p-2$ and the proof is complete. $\hfill\Box$
%\end{proof}


\begin{prpen}
Let $K_{n_{1},n_{2},\ldots,n_{p}}$ be the complete $p$-partite graph such that
${p\geq3}$ and \linebreak $3\leq n_{1}\leq n_{2}\leq\ldots\leq n_{p}$. Then $r_{rR}%
(K_{n_{1},n_{2},\ldots,n_{p}})=n_{1}-2$.
\end{prpen}
\proofen
Let $G=K_{n_{1},n_{2},\ldots,n_{p}}$ and
$X_{1}=\{x_{1},$ $\dots,$ $x_{n_{1}}\}$, $X_{2}=\{y_{1},\dots,y_{n_{2}}%
\},\dots,X_{p}$ be the {partite sets }of $G$. Let $F$ be {an} $r_{rR}(G)$-set.
{By Proposition \ref{o3}-(d) we deduce that }$\gamma_{rR}%
(G+F)\in\{2,3\}$, {and by }Propositions \ref{n}-{(a) and \ref{pp1}}
we must have $$\Delta(G+F)\geq n_{1}+\dots+n_{p}-2$$ implying that $|F|\geq
n_{1}-2$. On the other hand, the function~$g$ defined by $f(x_{1})=2,$ $f(x_{2})=1$ and $f(x)=0$ otherwise, is {an} RRD-function of $G+\{x_{i}%
x_{1}\mid3\leq i\leq n_{1}\}$ {yielding }${r}_{rR}(G)\leq|F|=n_{1}-2$.
Consequently, $r_{rR}(G)=n_{1}-2$. \hfill$\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Graphs {with }small restrained Roman reinforcement number}

In this section, we study graphs with small restrained Roman reinforcement
number. We begin with the following lemma.
\begin{lemen}
\label{nn}If $G$ is a connected graph of order $n\geq3$ with $\gamma
_{rR}(G)=n$, then $r_{rR}(G)=1$.
\end{lemen}
\proofen
By Proposition \ref{n}, $G\simeq C_{4},C_{5}%
,B_{p},B_{p,q}$ or $G$ is a tree with $\mathrm{diam}(G)\le5$. If ${G\in \{C_{4},C_{5}\}}$, then the {desired result }follows from Proposition
\ref{cn1}. If $G\in\{B_{p},B_{p,q}\}$, then the function $g$ defined by
$g(u_{1})=2,$ $g(u_{2})=g(u_{5})=0$ and $g(x)=1$ otherwise, is an RRD-function
of $G+u_{2}u_{5}$ and hence $r_{rR}(G)=1$. Hence, we assume that $G$ is a tree
with diameter at most $5$.

 Let $v_{1}v_{2}\dots v_{k}\;(k\ge3)$ be a diametral
path in $G$. Define the function $f$ by $f(v_{1})=f(v_{3})=0,$ $f(v_{2})=2$ and
$f(x)=1$ for the remaining vertices. Clearly, $f$ is an RRD-function of
$G+v_{1}v_{3}$ and hence $r_{rR}(G)=1$. \hfill$\square$


\begin{prpen}
\label{aaa}Let $G$ be a connected graph of order $n\geq4$ with $\gamma
_{rI}(G)\geq3.$ {If }$f=(V_{0},V_{1},V_{2})$ {is }a $\gamma_{rR}(G)$-function
with $V_{1}\neq\emptyset$, {then }
\[
r_{rR}(G)=1.
\]
\end{prpen}
\proofen
Let $f=(V_{0},V_{1},V_{2})$ be a $\gamma_{rR}%
(G)$-function {such that }$V_{1}\neq\emptyset$. If $\gamma
_{rR}(G)=n$, then the desired result comes from Lemma \ref{nn}.

Hence assume
that $\gamma_{rR}(G)<n$. Then $V_{0}\neq\emptyset$ and so $V_{2}\neq\emptyset
$. Since $G$ is connected and $V_{1}\neq\emptyset$, there exists a vertex
$w\in V_{1}$ such that $w$ is dominated by $V_{0}\cup V_{2}$. Note
that if $w$ has a neighbor in $V_{0}$ and another one
in $V_{2},$ then reassigning $w$ provides an
RRD-function with weight $\gamma_{rR}(G)-1,$ a contradiction.

Now,
if $w$ is adjacent to a vertex in $V_{2}$, {then }the function $g$ defined by
$g(w)=0$ and $g(x)=f(x)$ otherwise, is an RRD-function of $G+wz$ where $z\in
V_{0}$, of weight less than $\gamma_{rR}(G),$ {and thus }$r_{rR}(G)=1$. If $w$
is adjacent to a vertex in $V_{0}$, then the function $g$ defined by $g(w)=0$
and $g(x)=f(x)$ otherwise, is an RRD-function of $G+wu$ where $u\in V_{2}$, of
weight less than $\gamma_{rR}(G)$ and so $r_{rR}(G)=1$. This completes the
proof. \hfill$\square$

\begin{prpen}
Let $G$ be a connected graph of order $n$ with $\gamma_{rR}(G)\geq3$. Then
$r_{rR}(G)=1$ if and only if $\gamma_{rR}(G)=n$ or $G$ {has a function
}$f=(V_{0},V_{1},V_{2})$ of weight less than $\gamma_{rR}(G)$ such that one of
the following conditions holds\textup:
\begin{itemize}
\item[\emph{(i)}] $G[V_{0}]$ has at most two isolated vertices and $V_{2}$ dominates
$V_{0};$
\item[\emph{(ii)}] $G[V_{0}]$ has no isolated vertices and there is exactly one
vertex $v\in V_{0}$ which is not dominated by $V_{2}$.
\end{itemize}
\end{prpen}
\proofen
If $\gamma_{rR}(G)=n$, then by Lemma \ref{nn} we have
$r_{rR}(G)=1$. Hence suppose that ${\gamma_{rR}(G)<n}$, and let $f=(V_{0}%
,V_{1},V_{2})$ be a function on $G$ with weight less than $\gamma_{rR}(G)$
satisfying (i) or (ii). Since $\omega(f)<\gamma_{rR}(G)\leq n-2$, we have
$|V_{0}|\geq2$. In the case $V_{2}$ is non-empty,
let $u\in V_{2}$. Now, if (ii) holds, then $V_{2}\neq\emptyset$ and $f$ is
an RRD-function of $G+uv$.

 Assume now that (i) holds. If $G[V_{0}]$ has
two isolated vertices $w,v$, then $f$ is an RRD-function of $G+\{wv\}$ and
if $G[V_{0}]$ has exactly one isolated vertex, say $w$, then $f$ is an
RRD-function of $G+\{wz\}$, where $z$ is any vertex in $V_{0}-\{w\}$. Hence
in either case $r_{rR}(G)=1$.

Conversely, let $r_{rR}(G)=1$ and suppose that $\{uv\}$ is an $r_{rR}%
(G)$-set. If $\gamma_{rR}(G)=n$, then we are done. Hence suppose that
$\gamma_{rR}(G)\leq n-1$ and let $f$ be a $\gamma_{rR}(G+\{uv\})$-function.
{Notice that vertices }$u$ and $v$ {cannot be assigned both positive values
under }$f$ {(otherwise }$f$ is an RRD-function of~$G$). Without loss of
generality, assume that $f(u)=0$. If $f(v)=0$, then $f$ is a
function satisfying~(i). Hence assume that $f(v)\geq1$. If $u$ is adjacent to
a vertex with label~$2$ other than~$v$, then $f$ {is an RRD-function of }$G$.
Hence $u$ is not dominated by $V_{2}$ in $G$ and so $f$ is a
function satisfying~(ii). This completes the proof.\hfill$\square$


%\begin{proposition}
%\label{v2}Let $G$ be a connected graph of order $n$ with $\gamma
%_{rR}(G)\geq3$ {having }a $\gamma_{rR}(G)$-function $f=(V_{0},V_{1},V_{2})$
%such that $|V_{1}|=1$. Then $r_{rR}(G)\leq1$.
%\end{proposition}
%%\begin{proof}
%\noindent\textbf{Proof.} If $\gamma_{rR}(G)=n$, then the desired result comes from Lemma \ref{nn}. Thus suppose that $\gamma_{rR}(G)<n.$ {Then }$V_{0}%
%\neq\emptyset.$ {Let }$f$ be a $\gamma_{rR}(G)$-function such that $|V_{1}%
%|=1$. Let $V_{1}^{f}=\{v\}$ and let $N(v)=\{v_{1},\dots,v_{t}\}$. Notice that
%because of $|V_{1}|=1,$ for each $i$, {either }$f(v_{i})=2$ or $v_{i}$ has a
%neighbor {assigned }2. If $f(v_{i})=2$ for some $i$, then {no edge join }$v$
%to $V_{0}$ {and thus for any } $w\in V_{0}$ the function $g$ defined on
%$G+\{vw\}$ by $g(v)=0$ and $g(x)=f(x)$ otherwise, is {an }RRD-function of
%$G+\{wv\}$ of weight $\omega(f)-1$. {Hence we can assume that no }$v_{i}$ {is
%assigned 2. In that case, for any }$w\in V_{2}$ the function $g$ defined on
%$G+\{vw\}$ by $g(v)=0$ and $g(x)=f(x)$ otherwise, is {an }RRD-function of
%$G+\{wv\}$ of weight $\omega(f)-1$. Therefore, {in either case} we have
%$r_{rR}(G)=1$. $\hfill\Box$
%%\end{proof}


\begin{prpen}
Let $G$ be a connected graph of order $n$ with $\gamma_{rR}(G)\geq3$. If
$\delta(G)=1$, then {$r_{rR}(G)=1$.}
\end{prpen}
\proofen
{First note that $n\geq3,$
since }$\gamma_{rR}(G)\geq3.$ {If }$\gamma_{rR}(G)=n$, then the
result comes from Lemma \ref{nn}. Hence we assume that $\gamma_{rR}(G)<n$,
{and let }$f=(V_{0},V_{1},V_{2})$ be a $\gamma_{rR}(G)$-function.
{We have }$V_{0}\neq\emptyset$ {(because $\gamma_{rR}(G)<n$)
and thus }$V_{2}\neq\emptyset$.

 Let $u$ be a support vertex of $G$
and $u_{1}$ {a leaf }neighbor of $u$. By definition we have
$f(u_{1})\geq1$. If $f(u_{1})=1$ or $f(u)=1$, then the desired result comes
from Proposition \ref{aaa}.

Hence we assume that $f(u_{1})=2$ and
$f(u)\neq1$. The minimality of $f$ implies that $f(u)=0$.
Note that $u_{1}$ is the only neighbor of~$u$
which assigned a $2$, for otherwise~$u_{1}$ can be
reassigned the value~$1$ instead of~$2$. Let $w$ be a neighbor of $u$ with label~$0$. To Roman dominate $w$, there is a vertex $v$ such that $f(v)=2$. Then the
function $g$ defined on $G+uv$ by $g(u_{1})=1$ and $g(x)=f(x)$ otherwise, is
{an} RRD-function of $G+\{uv\}$ with weight $\omega(f)-1$. {Consequently,
}$r_{rR}(G)=1$.\hfill$\square$


%\begin{proposition}
%Let $G$ be a connected graph of order $n\geq3$ with $\gamma_{rI}(G)\ge3$. If
%$G$ has at least two leaves, then
%\[
%r_{rI}(G)\le2.
%\]
%This bound is sharp for a double star $DS_{p,q}\;(q\ge p\ge3)$.
%\end{proposition}
%%\begin{proof}
%\noindent\textbf{Proof.} If $\gamma_{rI}(G)=n$, then the result {follows from
%}Lemma \ref{nn}. Hence we assume that $\gamma_{rI}(G)<n$, {and let }%
%$f=(V_{0},V_{1},V_{2})$ be a $\gamma_{rI}(G)$-function. {Clearly, }$V_{0}%
%\neq\emptyset$. {Let }$w\in V_{0}$, {and let }$u_{1}$ and $v_{1}$ be two
%leaves of $G$ {whose support vertices are }$u$ and $v,$ respectively (possibly
%$u=v$). By definition $f(u_{1}),f(v_{1})\geq1$. If $f(u)\geq1$ (the case
%$f(v)\geq1$ is similar), then the function $g$ defined on $G+\{u_{1}%
%v_{1},u_{1}w\}$ by $g(u_{1})=0$ and $g(x)=f(x)$ otherwise, is {an
%}RID-function of $G+\{u_{1}v_{1},u_{1}w\}$ of weight $\omega(f)-1$. Hence,
%assume that $f(u)=f(v)=0$. Since $G$ is not a star, we have $(V_{1}\cup
%V_{2})-\{u_{1},v_{1}\}\neq\emptyset$. Let $z\in(V_{1}\cup V_{2})-\{u_{1}%
%,v_{1}\}$. Define the function $g$ on $G+\{u_{1}z,v_{1}z\}$ by $g(z)=\min
%\{2,f(z)+1\}=2$, $g(u_{1})=g(v_{1})=0$ and $g(x)=f(x)$ otherwise. Clearly, $g$
%is {an }RID-function of $G+\{u_{1}z,v_{1}z\}$ with weight $\omega(f)-1$.
%{Therefore, in either case we have} $r_{rI}(G)\leq2$. $\hfill\Box$


\begin{coren}
For any tree $T$ of order $n\geq3$, $r_{rR}(T)=1.$
\end{coren}

\section{Bounds on $r_{rR}(G)$}
In this section, we present some sharp upper bounds {on the
restrained Roman reinforcement number of a graph. }Given a set $S\subseteq V$
of vertices in a graph $G$ and a vertex $v\in S$, the \textit{external private
neighborhood} of $v$ with respect to $S$ in the set $$\mathrm{epn}(v,S)=\big\{u\in
V-S\mid N(u)\cap S=\{v\}\big\}.$$

\begin{prpen}
\label{aaaaa}Let $G$ be a connected graph with $\gamma_{rI}(G)\geq3.$ If
$f=(V_{0},V_{1},V_{2})$ is a \linebreak $\gamma_{rR}(G)$-function with $V_{2}%
\neq\emptyset$, then
\[
r_{rR}(G)\leq\min\big\{|\mathrm{epn}(v,V_{2})\cap V_{0}|\;:\;v\in V_{2}\big\}.
\]
\end{prpen}
\proofen
Let $f=(V_{0},V_{1},V_{2})$ be a $\gamma_{rR}%
(G)$-function with $V_{2}\neq\emptyset$. If $|\mathrm{epn}(v,V_{2})\cap
V_{0}|=0$ for some vertex $v\in V_{2},$ then reassigning $v$ the value $1$
instead of $2$ provides an RRD-function of weight less than $\gamma_{rR}(G)$
leading to a contradiction. Hence $|\mathrm{epn}(v,V_{2})\cap V_{0}|\geq1$
for every $v\in V_{2}.$ Let $u$ be a vertex in $V_{2}$ such that
$$|\mathrm{epn}(u,V_{2})\cap V_{0}|=\min\{|\mathrm{epn}(v,V_{2})\cap V_{0}|:\;v\in V_{2}\}$$ and let $\mathrm{epn}(u,V_{2})\cap V_{0}%
=\{u_{1},\dots,u_{\epsilon}\}$. If $|V_{2}|\geq2$ and $w\in V_{2}-\{u\}$, then
the function $g$ defined by $g(u)=1$ and $g(x)=f(x)$ otherwise, is an
RRD-function of $G+\{wx\mid x\in\mathrm{epn}(v,V_{2})\cap V_{0}\}$ of weight
less than $\gamma_{rR}(G)$ and so $$r_{rR}(G)\leq\min\{|\mathrm{epn}
(v,V_{2})\cap V_{0}|:\;v\in V_{2}\}.$$ Hence assume that $V_{2}=\{u\}$. Then
$u$ dominates all vertices in $V_{0}$. Since $\gamma_{rR}(G)\geq3$, we have
$V_{1}\neq\emptyset$ and the desired result follows from
Proposition \ref{aaa}. \hfill$\square$\\[1ex]



We observe that for any $\gamma_{rR}(G)$-function $f=(V_{0},V_{1},V_{2})$,
every vertex $u$ of $V_{2}$ {can have }at most $d_{G}(u)$ neighbors in $V_{0}%
$. Whence we have the following corollary.

\begin{coren}
\label{V22}Let $G$ be a connected graph with ${\gamma_{rR}(G)\ge3}$ and
${f=(V_{0},V_{1},V_{2})}$ a \linebreak $\gamma_{rR}(G)$-function {with $|V_{2}|\ge1$}. Then
$r_{rR}(G)\leq\Delta.$
\end{coren}

%The bounds of Proposition \ref{V2} and Corollary \ref{V22} are sharp for the
%double star $DS_{r,r}$ with $r\ge2$.


\begin{coren}
Let $G$ be a connected graph with $\gamma_{rR}(G)\geq3$ containing a path
$v_{1}v_{2}v_{3}v_{4}v_{5}$ in which $d_{G}(v_{i})=2$ for $i\in\{2,3,4\}$.
Then $r_{rR}(G)\leq2$.
\end{coren}

\proofen
If $\gamma_{rR}(G)=n$, then the result is immediate
{from }Lemma \ref{nn}. Hence we assume that $\gamma_{rR}(G)<n$, {and let
}$f=(V_{0},V_{1},V_{2})$ be a $\gamma_{rR}(G)$-function. By Proposition
\ref{aaa}, we may assume that $V_{1}=\emptyset$. Then we must have
$2\in\{f(v_{2}),f(v_{3}),f(v_{4})\}$ and {the result }follows from Proposition
\ref{aaaaa}. \hfill$\square$\\[1ex]


%\begin{proposition}
%\label{V1} Let $G$ be a connected graph of order $n\geq4$ with $\gamma
%_{rI}(G)\geq3$ and let $f=(V_{0},V_{1},V_{2})$ a $\gamma_{rI}(G)$-function
%with $V_{1}\neq\emptyset$. Then
%\[
%r_{rI}(G)\leq2+\min\{\epsilon(v)\mid v\in V_{1}\}.
%\]
%\end{proposition}
%%\begin{proof}
%\noindent\textbf{Proof.} {Let }$f=(V_{0},V_{1},V_{2})$ be a $\gamma_{rI}%
%(G)$-function with $V_{1}\neq\emptyset$. If $\gamma_{rI}(G)=n$, then the
%result {follows from} Lemma \ref{nn}. Hence we assume that $\gamma_{rI}(G)<n$.
%{Clearly, }$V_{0}\neq\emptyset$. {Let }$v\in V_{0}$, {and let }$u$ be a vertex
%in $V_{1}$ such that $\epsilon(u)=\min\{\epsilon(w)\mid w\in V_{1}\}.$ {Let
%}$N_{\epsilon}(u)=\{u_{1},\dots,u_{\epsilon}\}$ when $\epsilon(u)\geq1$ {and
%let }$V_{1}-\{u\}=\{z_{1},\dots,z_{k}\}$.
%First {assume that }$N(u)\cap V_{0}=\emptyset$. If $V_{2}\neq\emptyset$ and
%$w\in V_{2}$, then {consider }the function $g$ defined by $g(u)=0$ and
%$g(x)=f(x)$ otherwise. {One can check that }$g$ {has weight }$\omega(f)-1$
%{and it is an RID-function on the graph }$G^{\prime}$ {obtained from }$G$ {by
%adding the edge }$vu$ {and possibly the edge }$uw$ {if }$uw\notin E(G).$ Hence
%we can assume that $V_{2}=\emptyset$. {Now, if }$\sum_{x\in N(u)}f(x)\geq2$,
%then the function $g$ {defined on }$G+\{vu\}$ by $g(u)=0$ and $g(x)=f(x)$
%otherwise, is {an } RID-function of weight $\omega(f)-1$. {Moreover, if }%
%$\sum_{x\in N(u)}f(x)=0$, then the function $g$ {defined }on $G+\{z_{1}%
%u,z_{2}u\}$ by $g(u)=0$ and $g(x)=f(x)$ otherwise, is {an} RID-function of
%weight $\omega(f)-1$. {Finally, if }$\sum_{x\in N(u)}f(x)=1$, {then s}ince
%$\gamma_{rI}(G)\geq3$, there is a vertex $z\in V-N[u]$ with $f(z)=1$. {In this
%case, the function }$g$ {defined }on $G+\{vu,zu\}$ by $g(u)=0$ and $g(x)=f(x)$
%otherwise, {is an }RID-function of weight $\omega(f)-1$. {All of the above
%situations lead to }$r_{rI}(G)\leq2,$ {and the result is valid. }
%{For the sequel, we can assume that }$N(u)\cap V_{0}\neq\emptyset$. If
%$V_{2}\neq\emptyset$ and $w\in V_{2}$, then the function $g$ {defined }on
%$G_{1}=G+\{wu_{1},\dots,wu_{\epsilon},wu\}$ by $g(u)=0$ and $g(x)=f(x)$
%otherwise, is {an} RID-function of $G_{1}$ of weight $\omega(f)-1$. Hence
%assume that $V_{2}=\emptyset$. Since $\gamma_{rI}(G)\geq3$, we have
%$|V_{1}|\geq3$ and {thus }for any vertex $u_{i}$, there exists a vertex
%$u_{i}^{\prime}\in V_{1}$ such that $u_{i}u_{i}^{\prime}\not \in E(G)$. {Now,
%if }$\sum_{x\in N(u)}f(x)\geq2$, then the function $g$ on $G_{2}%
%=G+\{u_{i}u_{i}^{\prime}\mid1\leq i\leq\epsilon(u)\}$ defined by $g(u)=0$ and
%$g(x)=f(x)$ otherwise, is {an} RID-function of $G_{2}$ of weight $\omega
%(f)-1$. If $\sum_{x\in N(u)}f(x)=0$, then the function $g$ on $G_{3}%
%=G+\{z_{1}u,z_{2}u,u_{i}u_{i}^{\prime}\mid1\leq i\leq\epsilon(u)\}$ by
%$g(u)=0$ and $g(x)=f(x)$ otherwise, is {an} RID-function of $G_{3}$ of weight
%$\omega(f)-1$. {Finally, assume that }$\sum_{x\in N(u)}f(x)=1$. {Without loss
%of generality, assume that }$z_{2}\not \in N[u]$ {and let }$F=\{z_{2}%
%u,u_{i}u_{i}^{\prime}\mid1\leq i\leq\epsilon(u)\}.$ {By defining the function
%}$g$ on $G+F$ as before, {we can see that }$g$ is {an }RID-function of $G+F$
%of weight $\omega(f)-1$. {Consequently, in either case }we get $r_{rI}%
%(G)\leq\epsilon(u)+2=2+\min\{\epsilon(v)\mid v\in V_{1}\}$. This completes the
%proof. $\hfill\Box$
%%\end{proof}
\begin{figure}[ht]
\centering
\begin{tikzpicture}%[scale=.3, transform shape]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%(1)
\node [draw, shape=circle,fill=black,scale=0.3] (u) at (10.5,4.5){};
%\node [draw, shape=circle,fill=black,scale=0.3] (u0) at (12.5,4.5){};
\node [draw, shape=circle,fill=white,scale=0.3] (u1) at (7,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (u2) at (8,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (v1) at (9,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (v2) at (10,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (u11) at (11,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (u21) at (12,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (v11) at (13,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (v21) at (14,2.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1u1) at (7,1.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1u2) at (8,1.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1v1) at (9,1.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1v2) at (10,1.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1u11) at (11,1.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1u21) at (12,1.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1v11) at (13,1.7){};
\node [draw, shape=circle,fill=white,scale=0.3] (1v21) at (14,1.7){};
%\node [draw, shape=circle,fill=white,scale=0.3] (w1) at (8.5,1.7){};
%\node [draw, shape=circle,fill=white,scale=0.3] (w2) at (8.5,.8){};
\node [draw, shape=circle,fill=black,scale=0.3] (v) at (10.5,0.2){};
%\node [draw, shape=circle,fill=black,scale=0.3] (w) at (12.5,0.2){};
%\node [draw, shape=circle,fill=white,scale=0.3] (y5) at (10,1){};
%\node [draw, shape=circle,fill=white,scale=0.3] (y6) at (11,1){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v1) at (3.2,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v2) at (4.1,-5.6){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v3) at (4.1,-6.4){};
%\node [draw, shape=circle,fill=black,scale=0.3] (v4) at (3.2,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v5) at (2.3,-6.4){};
%\node [draw, shape=circle,fill=white,scale=0.3] (v6) at (2.3,-5.6){};
%\draw  [line width=0.8pt](x1)--(y1)--(x2)--(y4)--(x3)--(y6)--(x4)--(y5)--(y6);
%\draw  [line width=0.8pt](u)--(u1)--(v)--(u2)--(u);
%\draw  [line width=0.8pt](w)--(v11)--(v21)--(w)--(u11)--(u21)--(w);
%\draw  [line width=0.8pt](u0)--(v1)--(v2)--(u0)--(u1)--(u2)--(u0)--(u11)--(u21)--(u)--(v11)--(v21)--(u0);
\draw  [line width=0.8pt](u)--(v1)--(v2)--(u)--(u1)--(u2)--(u)--(u11)--(u21)--(u)--(v11)--(v21)--(u);
\draw  [line width=0.8pt](v)--(1v1)--(1v2)--(v)--(1u1)--(1u2)--(v)--(1u11)--(1u21)--(v)--(1v11)--(1v21)--(v);
%\draw  [line width=0.8pt](w)--(1v1)--(1v2)--(w)--(1u1)--(1u2)--(w)--(1u11)--(1u21)--(w)--(1v11)--(1v21)--(w);
\draw  [line width=0.8pt](u2)--(1u2);
\draw  [line width=0.8pt](v2)--(1v2);
\draw  [line width=0.8pt](u1)--(1u1);
\draw  [line width=0.8pt](v1)--(1v1);
\draw  [line width=0.8pt](u21)--(1u21);
\draw  [line width=0.8pt](v21)--(1v21);
\draw  [line width=0.8pt](u11)--(1u11);
\draw  [line width=0.8pt](v11)--(1v11);
%\draw  [line width=0.8pt](u0)--(u11)--(u21)--(u0)--(v11)--(v21)--(u0);
%\draw  [line width=0.8pt](v1)--(v2);
%\draw  [line width=0.8pt](w1)--(w2);
%\node at (7,3) {$u_1$};
%\node at (7.8,3) {$u_2$};
%\node at (9,2.9) {$v_1$};
%\node at (9.8,2.9) {$v_2$};
\node at (10.5,4.7) {$x$};
%\node at (12.5,4.7) {$y$};
\node at (10.5,-.1) {$y$};
%\node at (12.5,0) {$w$};
%\node at (8.5,.5) {$w_1$};
%\node at (8.5,2) {$w_2$};
%\node at (11,.7) {$y_6$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture}
\caption{A graph $G$ of order 18 and $r_{rR}(G)=4$. }%
\label{FAZ23333}%
\end{figure}

\bigskip

Using Propositions \ref{aaa} and \ref{aaaaa} we {obtain }the next result.

\begin{theoen}
\label{n/2}For any graph $G$ of order $n\geq3$, we have \ {%
\[
r_{rR}(G)\leq\max\{1,(2n-\gamma_{rR}(G))/\gamma_{rR}(G)\}.
\]
Moreover, the bound is sharp.}
\end{theoen}

\proofen
If $\gamma_{rR}(G)=2$, then $r_{rR}(G)=0$ and the
result is true.

If $\gamma_{rR}(G)=n$, then by Lemma~ \ref{nn}, $r_{rR}(G)=1$
and the desired result follows.

Hence we assume that $3\leq\gamma
_{rR}(G)<n$, and let $f=(V_{0},V_{1},V_{2})$ be a $\gamma_{rR}(G)$-function.
If $V_{1}\neq\emptyset$, then the result {follows }from Proposition
\ref{aaa}. {Thus suppose }that $V_{1}=\emptyset$. Then
$\gamma_{rR}(G)/2=|V_{2}|\geq2$ and clearly $$|\mathrm{epn} (u,V_{2})\cap V_{0}|\leq (2n-\gamma_{rR}(G))/\gamma_{rR}(G)$$ for some
$u\in V_{2}$. Now, the result is immediate by Proposition \ref{aaa}.

To show the sharpness, consider the graph $G$ illustrated in Figure
\ref{FAZ23333}. It is easy to see that $\gamma_{rR}(G)=4$ and the function $f$
on $G$ defined by $f(x)=f(y)=2$ and $f(z)=0$ otherwise, is {the }unique
$\gamma_{rR}(G)$-function. Then $$r_{rR}(G)\leq(2n-\gamma_{rR}(G))/\gamma_{rR}(G)=8.$$ Now let $F$ be {an} $r_{rR}(G)$-set. Then $\gamma
_{rR}(G+F)\leq3$ and so $\Delta(G+F)\geq n-2$ (see Propositions~\ref{n}-{(a) and \ref{pp1}}). This implies that $|F|\geq8$, and
consequently, $$r_{rR}(G)=8=(2n-\gamma_{rR}(G))/\gamma_{rR}(G).$$
\hfill$\square$
\section{Conclusion}
{The main objective of this paper was to start the study of the restrained
Roman reinforcement number $r_{rR}(G)$ of a graph $G.$ We first showed that
the decision problem associated with the restrained Roman reinforcement
problem is NP-hard, and then various properties as well as some
sharp bounds of the restrained Roman reinforcement number have
been established. In particular we showed that $r_{rR}(T)=1$
for every tree $T$ of order at least three and that
$r_{rR}(G)\leq\Delta(G)$ for any connected graph $G$ with $\gamma_{rR}%
(G)\geq3$. As a future work, one can focus on the problem of characterizing
all connected graphs $G$ such that $r_{rR}(G)=\Delta(G)$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
This research was financially supported by fund from Hubei Province Key Laboratory of Intelligent Information Processing and Real-time Industrial System (Wuhan University of Science and Technology).

\begin{Biblioen}                                                                                            %
\bibitem{AAC}
{Abdollahzadeh Ahangar~H.,  Amjadi~J.,  Chellali~M., Nazari-Moghaddam~S.,  Sheikholeslami~S.\,M.} Total Roman reinforcement in graphs.
\emph{Discuss. Math. Graph Theory,} 2019. Vol.~39, No.~4. P.~787--803. \hrefdoi{10.7151/dmgt.2108}

\bibitem {AAC1}
{Abdollahzadeh Ahangar~H.,  Mirmehdipour~S.\,R.}
 Bounds on the restrained Roman domination number of a graph.
\emph{Commun. Comb. Optim., } 2016. Vol.~1, No.~1. P.~75--82.
\hrefdoi{10.22049/CCO.2016.13556}

\bibitem {ADF}
{ Amjadi~J.,  Asgharsharghi~L.,  Dehgardi~N.,  Furuya~M.,
Sheikholeslami~S.\,M. and  Volkmann~L.}
The $k$-rainbow reinforcement numbers in graphs.
\emph{Discrete Appl. Math.,} 2017. Vol.~217. P.~394--404.
\hrefdoi{10.1016/j.dam.2016.09.043}

\bibitem {amjadi}
{Amjadi~J.,  Sadeghi~H.}
Double Roman reinforcement number in graphs.
\emph{AKCE Int. J. Graphs Comb.,} 2021. Vol.~18, No.~3. P.~191--199.
\hrefdoi{10.1080/09728600.2021.1997557}

\bibitem{EACS}
{Ebrahimi~N., Amjadi~J., Chellali~M., Sheikholeslami~S.M.}
Quasi-total Roman reinforcement in graphs.
\emph{AKCE Int. J. Graphs Comb.,} 2022. In press.
\hrefdoi{10.1080/09728600.2022.2158051}


\bibitem {GJ}
{Garey~M.\,R., Johnson~D.\,S.}
\emph{Computers and Intractability: A Guide to the Theory of NP-Completness.}
Freeman:~San Francisco, 1979. 340~p.

\bibitem {HSS}
{Hao~G.,  Sheikholeslami~S.\,M.,  Wei~S.}
Italian reinforcement number in graphs.
\emph{IEEE Access}, 2019. Vol.~7. Art.~no.~184448.
\hrefdoi{10.1109/ACCESS.2019.2960390}

\bibitem {hhs}
{ Haynes~T.\,W.,  Hedetniemi~S.\,T.,  Slater~P.\,J.}
\emph{Fundamentals of Domination in Graphs.} Marcel Dekker, Inc., New York,
1998. 446~p.
\hrefdoi{10.1201/9781482246582}

\bibitem {JS}
{Jafari Rad~N.,  Sheikholeslami~S.\;M.}
Roman reinforcement in graphs.
\emph{Bull. Inst. Combin. Appl.,} 2011. Vol.~61. P.~81--90.

\bibitem {KM}
{ Kok~J.,  Mynhardt~C.\,M.}
Reinforcement in graphs.
\emph{Congr. Numer.,} 1990. Vol.~79. P.~225--231.

\bibitem {PP}
{Pushpam~P.\,R.\,L.,  Padmapriea~S.}
Restrained Roman domination in graphs.
\emph{Trans. Comb.,} 2015. Vol.~4, No.~1. P.~1--17.
\hrefdoi{10.22108/TOC.2015.4395}

\bibitem{samadi}
{Samadi~B.,  Soltankhah~N., Abdollahzadeh Ahangar~H., Chellali~M., Mojdeh~D.\,A.,  Sheikholeslami~S.\,M.,  Valenzuela-Tripodoro~J.\,C.}
Restrained condition on double Roman dominating functions.
\emph{Appl. Math. Comput.,} 2023. Vol.~438. Art. no.~127554.
\hrefdoi{10.1016/j.amc.2022.127554}

\bibitem {SAKS}
{Shahbazi~L., Abdollahzadeh Ahangar~H., Khoeilar~R., Shekholeslami~S.M.}
Total $k$-rainbow reinforcement number in graphs.
\emph{Discrete Math. Algorithms Appl.,} 2021. Vol.~13, No.~1. Art.~no.~2050101.
\hrefdoi{10.1142/S1793830920501013}

\bibitem {AAC2}
{Siahpour~F., Abdollahzadeh Ahangar~H., Sheikholeslami~S.\,M.}
Some progress on the restrained Roman domination.
\emph{Bull. Malays. Math. Sci. Soc.,} 2021. Vol.~44, No.~7. P.~733--751.
\hrefdoi{10.1007/s40840-020-00965-0}

\bibitem{V1}
{Volkmann~L.}
Remarks on the restrained Italian domination number in graphs.
\emph{Commun. Comb. Optim., } 2023. Vol.~8, No.~1. P.~183--191.
\hrefdoi{10.22049/CCO.2021.27471.1269}

\bibitem{V2}
{Volkmann~L.}
Restrained double Italian domination in graphs.
\emph{Commun. Comb. Optim., } 2023. Vol.~8, No.~1. P.~1--11.
\hrefdoi{10.22049/CCO.2021.27334.1236}

\end{Biblioen}
\end{document}
