Suppose [3]={0,1,2,3}\documentclass[12pt]{minimal}
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\begin{document}$$[3]=\{0,1,2,3\}$$\end{document} and [3-]={-1,1,2,3}\documentclass[12pt]{minimal}
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\begin{document}$$[3^{-}]=\{-1,1,2,3\}$$\end{document}. An outer independent signed double Roman dominating function (OISDRDF) of a graph Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} is function l:V(Γ)→[3-]\documentclass[12pt]{minimal}
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\begin{document}$$l:V({\Gamma })\rightarrow [3^{-}]$$\end{document} for which (i) each vertex t with l(t)=-1\documentclass[12pt]{minimal}
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\begin{document}$$l(t)=-1$$\end{document} is joined to at least two vertices labeled a 2 or to at least one vertex z with l(z)=3\documentclass[12pt]{minimal}
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\begin{document}$$l(z)=3$$\end{document}, (ii) each vertex t with l(t)=1\documentclass[12pt]{minimal}
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\begin{document}$$l(t)=1$$\end{document} is joined to at least a vertex z with l(z)≥2,\documentclass[12pt]{minimal}
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\begin{document}$$l(z)\ge 2,$$\end{document} (iii) l(N[t])=∑w∈N[t]l(w)≥1\documentclass[12pt]{minimal}
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\begin{document}$$l(N[t])=\sum _{w\in N[t]}l(w)\ge 1$$\end{document} occurs for each vertex t, (iv) the set of vertices labeled -1\documentclass[12pt]{minimal}
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\begin{document}$$-1$$\end{document} 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) γsdRoi(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{sdR}^{oi}(\Gamma )$$\end{document} is the minimum weight of an OISDRDF on Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}. We first show that determining the number γsdRoi(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{sdR}^{oi}(\Gamma )$$\end{document} 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≥3,\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3,$$\end{document}γsdRoi(Γ)≤n-1,\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{sdR}^{oi}(\Gamma )\le n-1,$$\end{document} and we characterize extremal trees attaining this bound.
