Suppose \([3]=\{0,1,2,3\}\) and \([3^{-}]=\{-1,1,2,3\}\). An outer independent signed double Roman dominating function (OISDRDF) of a graph \(\Gamma \) is function \(l:V({\Gamma })\rightarrow [3^{-}]\) for which (i) each vertex t with \(l(t)=-1\) is joined to at least two vertices labeled a 2 or to at least one vertex z with \(l(z)=3\), (ii) each vertex t with \(l(t)=1\) is joined to at least a vertex z with \(l(z)\ge 2,\) (iii) \(l(N[t])=\sum _{w\in N[t]}l(w)\ge 1\) occurs for each vertex t, (iv) the set of vertices labeled \(-1\) under l is an independent set. The weight of an OISDRDF is the sum of its function values over all vertices, and the outer independent signed double Roman domination number (OISDRD-number) \(\gamma _{sdR}^{oi}(\Gamma )\) is the minimum weight of an OISDRDF on \(\Gamma \). We first show that determining the number \(\gamma _{sdR}^{oi}(\Gamma )\) is NP-complete for bipartite and chordal graphs. Then we provide exact values of this parameter for paths and cycles. Moreover, we show that for trees T of order \(n\ge 3,\) \(\gamma _{sdR}^{oi}(\Gamma )\le n-1,\) and we characterize extremal trees attaining this bound.

假设\（[3] = \ {0,1,2,3 \} \）和\（[3 ^ {-}] = \ {-1,1,2,3 \} \）。图\（\ Gamma \）的外部独立有符号双罗曼支配函数（OISDRDF）是函数\（l：V（{\ Gamma}）\ rightarrow [3 ^ {-}] \），每个（i）顶点吨与\（L（T）= - 1 \）被接合到一个标记为2的至少两个顶点，或者至少一个顶点ž与\（升（Z）= 3 \） ，（ⅱ）每个顶点吨与\（l（t）= 1 \）至少与一个具有\（l（z）\ ge 2 \\）（iii）的顶点z连接，（\ [l（N [t]）= \ sum _ {w \对于每个顶点t，在N [t]} l（w）\ ge 1 \）中发生，（iv）在l下标为\（-1 \）的顶点集合是一个独立集合。OISDRDF的权重是其所有顶点上的函数值的总和，外部独立有符号双罗马统治数（OISDRD-number）\（\ gamma _ {sdR} ^ {oi}（\ Gamma）\）是\（\ Gamma \）上的OISDRDF的最小权重。我们首先表明，对于二部图和弦图，确定数字\（\ gamma _ {sdR} ^ {oi}（\ Gamma）\）是NP完全的。然后，我们为路径和循环提供此参数的精确值。此外，我们证明对于树T的\（n \ ge 3，\）\（\ gamma _ {sdR} ^ {oi}（\ Gamma）\ le n-1，\） 并且我们描绘了达到此界限的极树。