The outer-independent signed Roman domination function (OISRDF) on \(\textit{G}\) =( \(\textit{V}\) , \(\textit{E}\) ) is the mapping \(\textit{f}\) : \(\textit{V}\rightarrow \) { \(\text {-1} {\textbf {, }}\text {1} {\textbf {, }}\text {2}\) }, which fulfills (i) for any \(\textit{w}\in \textit{V}\) , \(\textit{f}(\textit{N}[\textit{w}])=\sum _{\textit{z} \in \textit{N}[\textit{w}]} \textit{f}(\textit{z}) \ge \text {1}\) ; (ii) a vertex with function value \(\text {-1}\) is contiguous to a minimum of vertex with function value \(\text {2}\) ; (iii) all vertices labeled by \(\text {-1}\) is independent. The weight of OISRDF \(\textit{f}\) is \(\textit{w} {\textbf {(}} \textit{f} {\textbf {)}}=\sum _{\textit{v}\in \textit{V}}=-|\textit{V}_{\text {-1}}|+|\textit{V}_{\text {1}}|+\text {2}|\textit{V}_{\text {2}}|\) where \(\textit{V}_{\text {i}}\) ={ \(\textit{v}\in \textit{V} {\textbf {: }} \textit{f} {\textbf {(}} \textit{v} {\textbf {)}}={i}\) }, \({i}\in \) { \(\text {-1} {\textbf {, }}\text {1} {\textbf {, }}\text {2}\) }. The outer-independent signed Roman domination number \({\gamma ^{\textit{oi}}_{\textit{sR}} \textit{(G)}}\) is the minimal weight OISRDF \(\textit{f}\) of \(\textit{G}\) . Initially, we establish the relationship between \(|{\textit{V}_{\text {-1}}} |\) and \(|\textit{V}_\textit{2} |\) in a general graph. Furthermore, we provide some bounds for the OISRDF in paths, complete graphs, and complete bipartite graphs. Then, we demonstrate that the problem of OISRDF is NP-complete on bipartite and chordal graphs. Finally, we validate the upper bounds for the \({\gamma ^{\textit{oi}}_{\textit{sR}} \textit{(P}_{\textit{m}} \times \textit{P}_{\textit{n}} \textit{)}}\) of the Cartesian product graphs \(\textit{P}_{\textit{m}} \times \textit{P}_{\textit{n}}\) , leveraging the parity differences between \(\textit{m}\) and \(\textit{n}\) in the \(\textit{P}_{\textit{m}}\) and \(\textit{P}_{\textit{n}}\) .
