Let G be a connected graph of order at least three. A set \(D\subseteq V(G)\) is called a total outer-independent dominating set of G if every vertex of G is adjacent to at least one vertex in D, and \(V(G){\setminus } D\) is an independent set of G. The total outer-independent domination number of G, denoted by \(\gamma _t^{oi}(G)\) , is the minimum cardinality among all total outer-independent dominating sets of G. In this article, we study the total outer-independent domination number of subdivision graphs \(\texttt{S}(G)\) of connected graphs G. We establish several combinatorial results concerning this parameter in the context of this well-known graph operator, expressed in terms of some invariants of G, which improves for the case of trees. Finally, we derive closed formulas on \(\gamma _t^{oi}(\texttt{S}(G))\) for some well-known families of graphs G.
