a characterization of locating roman domination edge critical graphs

A roman dominating function (or just textit{rdf}) on a graph $g =(v, e)$ is a function $f: v longrightarrow {0, 1, 2}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. the weight of an textit{rdf} $f$ is the value $f(v)=sum_{u in {v}}f(u)$. an textit{rdf} $f$ can be represented as $f=(v_0,v_1,v_2)$, where $v_i={vin v:f(v)=i}$ for $i=0,1,2$. an textit{rdf} $f=(v_0,v_1,v_2)$ is called a locating roman dominating function (or just textit{ltextit{rdf}}) if $n(u)cap v_2neq n(v)cap v_2$ for any pair $u,v$ of distinct vertices of $v_0$. the locating-roman domination number $gamma_r^l(g)$ is the minimum weight of an textit{ltextit{rdf}} of $g$. a graph $g$ is said to be a locating roman domination edge  critical graph, or just $gamma_r^l$-edge critical graph, if $gamma_r^l(g-e)>gamma_r^l(g)$ for all $ein e$. the purpose of this paper is to characterize the class of $gamma_r^l$-edge critical graphs.