An independent double Roman dominating function (IDRDF) on a graph \(G=(V,E)\) is a function \(f{:}V(G)\rightarrow \{0,1,2,3\}\) having the property that if \(f(v)=0\), then the vertex v has at least two neighbors assigned 2 under f or one neighbor w assigned 3 under f, and if \(f(v)=1\), then there exists \(w\in N(v)\) with \(f(w)\ge 2\), such that the set of vertices with positive weight is independent. The weight of an IDRDF is the value \(\sum _{u\in V}f(u)\). The independent double Roman domination number \(i_\mathrm{dR}(G)\) of a graph G is the minimum weight of an IDRDF on G. We continue the study of the independent double Roman domination and show its relationships to both independent domination number (IDN) and independent Roman \(\{2\}\)-domination number (IR2DN). We present several sharp bounds on the IDRDN of a graph G in terms of the order of G, maximum degree and the minimum size of edge cover. Finally, we show that, any ordered pair (a, b) is realizable as the IDN and IDRDN of some non-trivial tree if and only if \(2a + 1 \le b \le 3a\).

中文翻译：

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图上的独立双罗马统治

图\（G =（V，E）\）上的独立双罗马统治函数（IDRDF）是函数\（f {：} V（G）\ rightarrow \ {0,1,2,3 \} \ ）具有，如果属性\（F（v）= 0 \） ，那么顶点v具有至少两个邻居指派下2 ˚F或一个相邻瓦特下分配3 ˚F，并且如果\（F（v）= 1 \ ），则存在\（w \ in N（v）\）与\（f（w）\ ge 2 \），因此具有正权重的顶点集是独立的。IDRDF的权重是值\（\ sum _ {u \ in V} f（u）\）。独立的双罗马统治数字\（i_ \ mathrm {dR}（G）\）图G的G是IDRDF在G上的最小权重。我们继续研究独立的双重罗马统治，并显示其与独立统治号码（IDN）和独立罗马\（\ {2 \} \）-统治号码（IR2DN）的关系。我们按照G的顺序，最大程度和边缘覆盖的最小大小在图G的IDRDN上给出了几个尖锐的边界。最后，我们证明，当且仅当\（2a + 1 \ le b \ le 3a \）时，任何有序对（a，  b）都可以实现为某些非平凡树的IDN和IDRDN 。