Constructive characterizations concerning total outer-independent domination in subdivision trees

Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set of vertices $D\subseteq V(G)$ 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 $G$. The subdivision graph of $G$, denoted by $\mathtt{S}(G)$, is the graph obtained from $G$ by subdividing every edge exactly once. Cabrera-Mart\'inez et al. [On the total outer-independent domination number of subdivision graphs, Comput. Appl. Math. 45 (2026) 315] proved that $\tfrac{4n(T)-l(T)-s(T)}{3}\leq \gamma_{t}^{oi}(\mathtt{S}(T))\leq \tfrac{4n(T)-l(T)+s(T)-2}{3}$ for any nontrivial tree $T$ of order $n(T)$ with $l(T)$ leaves and $s(T)$ support vertices. In this paper, we provide constructive characterizations of the families of trees that attain these bounds.