Critical properties on Roman domination graphs

A Roman domination function on a graph G is a function $r:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $r(u)=0$ is adjacent to at least one vertex $v$ for which $r(v)=2$. The weight of a Roman function is the value $r(V(G))=\sum_{u\in V(G)}r(u)$. The Roman domination number $\gamma_R(G)$ of $G$ is the minimum weight of a Roman domination function on $G$. "Roman Criticality" has been defined in general as the study of graphs where the Roman domination number decreases when removing an edge or a vertex of the graph. In this paper we give further results in this topic as well as the complete characterization of critical graphs that have Toman Domination number $\gamma_R(G)=4$.