AUTHOR = {Malini Mai, T.N.M. and Roushini Leely Pushpam, P.},
TITLE = {Weak Roman domination in graphs},
JOURNAL = {Discuss. Math. Graph Theory},
FJOURNAL = {Discussiones Mathematicae Graph Theory},
VOLUME = {31},
YEAR = {2011},
NUMBER = {1},
PAGES = {115-128},
ISSN = {1234-3099},
ABSTRACT = {Let $G=(V,E)$ be a graph and $f$ be a function $f:V\rightarrow\{0,1,2\}$. A
vertex $u$ with $f(u)=0$ is said to be \emph{undefended} with
respect to $f$, if it is not adjacent to a vertex with
positive weight. The function $f$ is a \emph{weak Roman
dominating function} (WRDF) if each vertex $u$ with $f(u)=0$
is adjacent to a vertex $v$ with $f(v)>0$ such that the
function $f^{'}: V \rightarrow \{0,1,2\}$ defined by
$f^{'}(u)=1, f^{'}(v)=f(v)-1$ and $f^{'}(w)=f(w)$ if $w\in
V-\{u,v\}$, has no undefended vertex. The \emph{ weight of }
$f$ is $w(f)=\sum_{v\in V}{f(v)}$. \emph{The weak
Roman domination number}, denoted by $\gamma_{r}(G)$, is the
minimum weight of a WRDF in $G$. In this paper, we
characterize the class of trees and split graphs for which
$\gamma_r(G)=\gamma(G)$ and find $\gamma_r$-value for a
caterpillar, a $2 \times n$ grid graph and a complete binary