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.

令\（G =（V，E）\）为简单图形。如果M中没有两个边具有相同的顶点，则集合\（M \ subseteq E \）是匹配项。的匹配数，表示为\（\β（G）\） （或\（\测试\） ），是在匹配的最大大小ģ。图G上的双罗曼支配函数（DRDF）是函数f：\（V \ longrightarrow \ {0,1,2,3 \} \）满足对于权重的每个顶点u （（f（u ）\ in \ left \ {0,1 \ right \} \）：\（\ left（i \ right）\）如果\（f（u）= 0 \），则u与至少一个具有\（f（v）= 3 \）的顶点v或两个顶点\（v_ {1} \），\（v_ {2} \）与\（f（v_ {1} ）= f（v_ {2}）= 2 \）。\（\左（ⅱ\右）\）如果\（F（U）= 1 \） ，然后ü邻近至少一个顶点v与\（F（V）\在\左\ {2,3 \对\} \）。双重罗马支配函数f的权重是值\（f（V）= \ sum _ {u \ in V} f（u）\）。上的曲线的双罗马控制函数的最小重量ģ被称为的双罗马控制数ģ，以\（\ gamma _ {dR} \ left（G \ right）\）表示。在本文中，首先，我们注意到\（\ gamma _ {dR}（G）\ le 3 \ beta（G）\），其中G是没有孤立顶点的图。此外，我们给出了满足\（\ gamma _ {dR}（G）= 3 \ beta（G）\）的框图G的描述性描述。最后，我们证明与\（\ gamma _ {dR}（G）= 3 \ beta（G）\）相关的决策问题对于二部图是\（CO- \数学{NP} \）-完全的。