Roman domination is a typical control problem with rich historical background and mathematical background, which is related to computer science, traffic safety supervision and control, enterprise safety production supervision and control, portfolio optimization, monitoring system and social network and other fields are closely related and have important theoretical significance and application value. The weak Roman domination number of graphs, denoted by $\gamma_{r}(G)$, is the minimum weight of a weak Roman dominating function in graphs. The domination number, denoted by $\gamma(G)$, is the minimum cardinality of a dominating set in $G$. We say that a graph $G$ is a weak Roman graph if $\gamma_{r}(G)=2\gamma(G)$. It is determined that the path $P_{3}$, stars $K_{1, t} (t\geq2)$ and trees $T$ which consist of the center vertices of stars $K_{1,t_{1}},K_{1, t_{2}},\cdots,K_{1, t_{n}}(t_{i}\geq3, i=1,2,\cdots,n)$ to form a path, or trees $T$ which made up of their outer vertices are weak Roman graph by means of construction, and some properties of weak Roman graphs and weak Roman dominating in graphs are given.