Extremal problems on weak Roman domination number

We show that the weak Roman domination number of a connected n-vertex graph is at most 2n3 and characterize the graphs achieving equality. In addition, we provide a constructive characterization of the trees for which the weak Roman domination number equals the Roman domination number and reveal several structural properties of these trees. This answers a problem posed in M. Chellali et al. (2014) [4].