Let k≥1\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 1$$\end{document} be an integer, and let D be a finite and simple digraph with vertex set V(D). A signed Roman k-dominating function (SRkDF) on a digraph D is a function f:V(D)→{-1,1,2}\documentclass[12pt]{minimal}
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\begin{document}$$f:V(D)\rightarrow \{-1,1,2\}$$\end{document} satisfying the conditions that (1) ∑x∈N-[v]f(x)≥k\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{x\in N^-[v]}f(x)\ge k$$\end{document} for each v∈V(D)\documentclass[12pt]{minimal}
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\begin{document}$$v\in V(D)$$\end{document}, where N-[v]\documentclass[12pt]{minimal}
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\begin{document}$$N^-[v]$$\end{document} consists of v and all vertices of D from which arcs go into v, and (2) every vertex u for which f(u)=-1\documentclass[12pt]{minimal}
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\begin{document}$$f(u)=-1$$\end{document} has an inner neighbor v for which f(v)=2\documentclass[12pt]{minimal}
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\begin{document}$$f(v)=2$$\end{document}. The weight of an SRkDF f is w(f)=∑v∈V(D)f(v)\documentclass[12pt]{minimal}
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\begin{document}$$w(f)=\sum _{v\in V(D)}f(v)$$\end{document}. The signed Roman k-domination number γsRk(D)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{sR}^k(D)$$\end{document} of D is the minimum weight of an SRkDF on D. In this paper we initiate the study of the signed Roman k-domination number of digraphs, and we present different bounds on γsRk(D)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{sR}^k(D)$$\end{document}. In addition, we determine the signed Roman k-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the signed Roman domination number γsR(D)=γsR1(D)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{sR}(D)=\gamma _{sR}^1(D)$$\end{document} and the signed Roman k-domination number γsRk(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{sR}^k(G)$$\end{document} of graphs G.
