@article{, 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 = {115128}, ISSN = {12343099}, 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 tree.}, }
