Let $$G=(V,E)$$ be a simple graph. A set $$M\subseteq E$$ is a matching if no two edges in M have a common vertex. The matching number, denoted $$\beta (G)$$ (or $$\beta $$ ), is the maximum size of a matching in G. A double Roman dominating function (DRDF) on a graph G is a function f: $$V\longrightarrow \{0,1,2,3\}$$ satisfying the conditions that for every vertex u of weight $$f(u)\in \left\{ 0,1\right\} $$ : $$\left( i\right) $$ if $$f(u)=0$$ , then u is adjacent to either at least one vertex v with $$f(v)=3$$ or two vertices $$v_{1}$$ , $$v_{2}$$ with $$f(v_{1})=f(v_{2})=2$$ . $$\left( ii\right) $$ if $$f(u)=1$$ , then u is adjacent to at least one vertex v with $$f(v)\in \left\{ 2,3\right\} $$ . The weight of a double Roman dominating function f is the value $$f(V)=\sum _{u\in V}f(u)$$ . The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number of G, denoted by $$\gamma _{dR}\left( G\right) $$ . In this paper, first, we note that $$\gamma _{dR}(G)\le 3\beta (G)$$ , where G is a graph without isolated vertices. Moreover, we give a descriptive characterization of block graphs G satisfying $$\gamma _{dR}(G)=3\beta (G)$$ . Finally, we show that the decision problem associated with $$\gamma _{dR}(G)=3\beta (G)$$ is $$CO-\mathcal {NP}$$ -complete for bipartite graphs.